From the Hele-Shaw Experiment to Integrable Systems: A Historical Overview
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From the Hele-Shaw Experiment to Integrable
Systems: A Historical Overview
Alexander Vasil’ev
to Björn Gustafsson on the occasion of his 60-th birthday
Abstract. This paper is a historical overview of the development of the topic
now commonly known as Laplacian Growth, from the original Hele-Shaw
experiment to the modern treatment based on the Hamiltonian formulation
of the contour evolution.
Mathematics Subject Classification (2000). Primary 76D7, 01A70; Secondary
30C35, 37K20.
Keywords. Hele-Shaw, Polubarinova-Kochina, Kufarev, Taylor, Saffman, Con-
formal Mapping, Integrable System.
1. Introduction
One of the most influential works in Fluid Dynamics at the end of the 19-th century
was a series of papers written by Henry Selby Hele-Shaw (1854–1941). There Hele-
Shaw first described his famous cell that became a subject of deep investigation
only more than 50 years later. A Hele-Shaw cell is a device for investigating two-
dimensional flow of a viscous fluid in a narrow gap between two parallel plates.
This cell is the simplest system in which multi-dimensional convection is present.
Probably the most important characteristic of flows in such a cell is that when
the Reynolds number based on gap width is sufficiently small, the Navier-Stokes
equations averaged over the gap reduce to a linear relation for the velocity similar
to Darcy’s law and then to a Laplace equation for the fluid pressure. Different
driving mechanisms can be considered, such as surface tension or external forces
(e.g., suction, injection). Through the similarity in the governing equations, Hele-
Shaw flows are particularly useful for visualization of saturated flows in porous
Supported by the grant of the Norwegian Research Council #177355/V30, and by the European
Science Foundation Research Networking Programme HCAA.
2 Alexander Vasil’ev
media, assuming they are slow enough to be governed by Darcy’s law. Nowadays,
the Hele-Shaw cell is used as a powerful tool in several fields of natural sciences
and engineering, in particular, matter physics, materials science, crystal growth
and, of course, fluid mechanics.
The modern treatment of the Hele-Shaw evolution is based on the Hamilton-
ian approach and on a general theory of plane contour motion, e.g., the Löwner
theory. A mathematical physics perspective, through integrable systems in partic-
ular, allows us to look at Hele-Shaw evolution as at a general contour dynamics in
the plane included into a dispersionless Toda hierarchy. What happened between
these two events? Who contributed to this topic?
This paper is my personal account on Hele-Shaw flows and on general con-
tour dynamics during the XX-th century from the original Hele-Shaw experiment
in 1897 to what is now commonly known as the Laplacian Growth problem, one of
the most challenging problems in non-linear physics. There were several persons
who influenced me and drew my attention to Hele-Shaw flows. One of them was
Yurii Hohlov who organized in 1996 a small seminar in Moscow. Dmitri Prokhorov,
Vladimir Gutlyanskiı̆, Konstantin Kornev, and me came to Moscow, and we dis-
cussed together with Yurii perspectives of the conformal mapping viewpoint on
the Hele-Shaw problem. It is worth mentioning that it was Hohlov who brought
Kufarev’s works to the Western audience and revealed the Soviet impact to the
development of Hele-Shaw flows. Another person who influenced me was Björn
Gustafsson whose 60-th birthday we celebrated recently, and I very much appre-
ciate his thorough treatment of weak solutions to the Hele-Shaw problem and his
potential-theoretic approach. 10 year’s work on Hele-Shaw flows and 4 years of my
collaboration with Björn resulted in the book [33] published in 2006. Of course, I
would also like to mention some earlier surveys covering certain topics or certain
time periods [47; 77; 103].
In 2007 Mark Mineev and Börn Gustafsson asked me to present a historical
overview of Hele-Shaw flows at the Banff International Research Station (Canada)
meeting. Working on that lecture I discovered many interesting and unknown (for
me) facts about persons who contributed to this interesting and challenging topic,
first of all about H. S. Hele-Shaw. After discussions with my colleagues at BIRS, I
decided to take a risk and to share this lecture with a wider audience adding some
information I was given during these discussions.
I do not pretend to cover all aspects of Hele-Shaw flows. Sam Howison and
Keith Gillow [25] collected more than 560 references on Hele-Shaw flows between
1897-1998. However, much more appeared during the last 10 years. A Google
search reveals more than 50 000 references on this topic. My much more modest
intent was to draw the reader’s attention to some interesting and bright persons
who were at the beginning of this boom and who sometimes became undeservedly
forgotten. Among the many documents referenced in this paper I distinguish an
informative obituary of Hele-Shaw, written by H. L. Guy [35], which I recommend
to an inquisitive reader for independent study.
From the Hele-Shaw Experiment to Integrable Systems 3
I would like to express my gratitude to many persons who influenced me, with
whom we discussed this topic, who gave me some information, my collaborators
and co-authors. Such a list of persons would occupy the rest of this paper so
let me keep them in my heart. Special thanks go to Linda Cummings and Björn
Gustafsson for their critical reading of the final version of this manuscript.
2. Hele-Shaw and his experiment
Hele-Shaw (1854–1941) was one of the most
prominent engineering researchers at the edge
of the XIX and XX centuries, a pioneer of tech-
nical education, a great organizer, President of
several engineering societies, including the Royal
Institution of Mechanical Engineers, Fellow of
the Royal Society, and sadly, an example of one
of the many undeservedly forgotten great names
in Science and Engineering.
Hele–Shaw was born on 29 July 1854 at
Billericay (Essex). The son of a successful solic-
itor Mr Shaw, he was a very religious person,
influenced by his mother from whom he adopted
her family name ‘Hele’ in his early twenties. At
Figure 1. H. S. Hele-Shaw
the age of 17 he finished a private education
and was apprenticed at the Mardyke Engineer-
ing Works, Messr Roach & Leaker in Bristol. His brother Philip E. Shaw (Lec-
turer and then Professor in Physics, University College Nottingham) testifies:
‘... Hele’s life from 17 to 24 was
a sustained epic: 10 hrs practical work
by day followed by night classes’. Hele-
Shaw applied for a 3 year Whitworth
Scholarship in Bristol and he was a lead-
ing candidate in the list before an exam,
when the congestion of lungs happened
and the effort and exposure would be
dangerous. Nevertheless, he went by cab
to the examination and again headed
Figure 2. Hele-Shaw’s birthplace the list and got the highest award of
£740. It is interesting that later in 1923
he founded the Whitworth Society.
2.1. 1876–1885
In 1876 he entered the University College Bristol (founded in 1872) and in 1878 he
was offered a position of Lecturer in Mathematics and Engineering under Professor
4 Alexander Vasil’ev
J. F. Main. In 1880 he got a Miller Scholarship from the Institution of Civil
Engineers for a paper on Small motive power.
In 1882 Main left the College and Hele-Shaw was appointed as Professor
of Engineering while the Chair in Mathematics was dropped. At that time he
organized his first Department of Engineering at the age of 27 and became its first
professor.
2.2. 1885–1904
In 1885 Hele-Shaw was invited to organize the Department of Engineering at the
University College Liverpool (founded in 1881), his second department, where he
served as a Professor of Engineering until 1904 when he moved to South Africa.
During this period Hele-Shaw carried out his seminal experiments at University
College Liverpool, designing the cell that bears his name.
2.3. 1904–1906
In 1904 Hele-Shaw became the first Pro-
fessor of Civil, Mechanical and Electri-
cal Engineering of the Transvaal Tech-
nical Institute (founded in 1903) which
then gave rise to the University of Jo-
hannesburg and the University of Pre-
toria.
It became his third department. In
1905 he was appointed Principal of the
Institute and an organizer of Technical
Figure 3. Transvaal Technical Insti- Education in the Transvaal. Hele-Shaw
tute thus became one of the pioneers of tech-
nical education not only in the metro-
politan area but also in the colonies.
Moreover, he was an exceptional teacher and his freehand drawing always
attracted special attention. He always tried to present difficult experiments in an
easier way, creating new devices in order to visualize certain phenomena.
2.4. 1906–1941
Upon returning from South Africa, Hele-Shaw abandoned academic life, setting
up as a consulting engineer in Westminster, concerning with development and
explotation of his own inventions. In 1920 Hele-Shaw became the Chairman of the
Educational Committee of the Institution of Mechanical Engineers, the British
engineering society, founded in 1847 by the Railway ‘father’ George Stephenson. In
1922 Hele-Shaw became the President of the Institution of Mechanical Engineers.
Hele-Shaw took a very active part in the professional and technical life of the
Great Britain. He was
• President of the Liverpool Engineering Society (1894);
• President of the Institution of Automobile Engineers (1909);
From the Hele-Shaw Experiment to Integrable Systems 5
Figure 4. Tay Bridge disaster
• President of the Association of Engineers in Charge (1912);
• President of Section G of the British Association for the Advancement of
Science (1915);
• President of the Institution of Mechanical Engineers (1922);
• Fellow of the Royal Society (1899).
One of his greatest contributions to Technical Education was the foundation of
‘National Certificates’ in Mechanical Engineering. He was joint Chairman of the
corresponding Committee (1920–1937).
Hele-Shaw was a man of great mental and physical alertness, of great energy
and of great courage. He was a self-made person and was successful and recognized
during his professional life. He possessed a great sense of humor, was a good
conversationalist (testimonies of his brother Philip, colleagues), loved companies.
He married Miss Ella Rathbone, a member of a prominent Liverpool family.
They had 2 children, the son was killed in combat during the I-st World War, the
daughter was married to Mr Harry Hall.
He retired at the age 85 from his office in London and died 1.5 year later on
30 January 1941.
2.5. Hele-Shaw’s inventions
Two of Hele-Shaw’s greatest inventions are his Stream-line Flow Methods (1896-
1900), and Automatic Variable-Pitch Propeller (1924), jointly with T. Beacham.
However, the full list of his inventions is much larger.
His earliest original work was reg-
istered in 1881 on the measurement of
wind velocity. At that time many en-
gineers tried to model the Tay Bridge
disaster (28 December 1879). Designed
by Thomas Bouch, the bridge over the
river Tay (near Dundee) was one of the
most remarkable engineering construc-
tions of that time taking six years to
build, and costing £300 000, as well as
Figure 5. Paris Motor Show a lot of constructive means and human
resources.
6 Alexander Vasil’ev
During a stormy night on 28 December 1879 the central sections of the bridge
collapsed taking with them a train of 6 carriages and 75 passengers, all of whom
perished. While modelling possible reasons for this disaster Hele-Shaw invented a
new integrating anemometer (a device to measure wind speed).
Continuing the list of his inventions let me mention a special stream-line filter
to purify water from oil pollution and the Hele-Shaw Friction Clutch (the first of
its kind; 1905) for cars, patent #GB795974. At a notable Paris Motor Show (1907)
about 80% of exhibited cars had the Hele-Shaw clutch.
His other inventions include the Hele-Shaw hydraulic transmission gear (1912),
Hele-Shaw pump (1923), and many others; a total of 82 patents.
Let us return to the most important inventions mentioned at the beginning
of this section. H. S. Hele-Shaw and T. E. Beacham patented the first constant
speed, variable pitch propeller in 1924, patent #GB250292. Later in 1929 Fairey
Figure 6. Variable pitch propellers
and Reed in the UK and Curtiss in the USA improved it and in 1932 variable
pitch propellers were introduced into air force service in both countries. Further
Figure 7. 1933 Boeing 247, passenger aircraft and 1935 Bristol Aeroplane
Company/Rolls-Royce: Bristol Type 130 Bombay, medium bomber
developments of the Hele-Shaw variable pitch propeller include:
• 1929 Adjustable pitch propeller drive, patent #GB1723617;
• 1931 Control system for propeller with controllable pitch, patent #GB1829930.
• 1932 Hele-Shaw and Beacham invented ‘Exactor Control’, a remote mecha-
nism to reproduce the control movements in aircrafts. Hele-Shaw was then
78 years old!
From the Hele-Shaw Experiment to Integrable Systems 7
The most notable result of Hele-Shaw’s scientific research came from his de-
sire to exhibit on a large screen the character of the flow past an object contained
in a lantern slide for students in Liverpool. He proposed a device consisting of
two parallel glass plates fixed at a small distance sandwiching a viscous fluid.
Hele-Shaw wanted to visualize stream lines of the flow. He tried to inject colour-
ing liquid (but it turned to be unsuitable, it immediately mixed with the rest
of fluid), and then sand (but it formed eddies, and then, modified the flow)...
He finally achieved his objective quite
by accident. Apparently the glass de-
veloped a small accidental leak provid-
ing small air bubbles acting as con-
tinuous tracers (1897). In 1897 Hele-
Shaw presented his method at the Royal
Institution of Naval Architects. Later
in 1898, Osborne Reynolds (1842–1912)
criticized the experiments by Hele-Shaw
Figure 8. Stream line method expecting turbulence at higher veloc-
ities. Indeed, O. Reynolds (1873) re-
vealed the turbulence phenomenon under higher velocities (his experiment is shown
in Figure 9 with sketches taken from the original Reynold’s paper).
Figure 9. Reynold’s experiment
Hele-Shaw’s greatest discovery in this context was that if the glass plates on
the lantern slide are mounted sufficiently close (0.02 inch) to each other, then the
flow is laminar at all velocities! He got the Gold Medal from the Royal Institution
of Naval Architects in 1898 for his stream line method.
8 Alexander Vasil’ev
Sir George Gabriel Stokes wrote:
‘Hele-Shaw’s experiments afford a com-
plete graphical solution, experimentally
obtained, of a problem which from its
complexity baffles mathematicians ex-
cept in a few simple cases’. Stokes men-
tions also Hele-Shaw’s experiments in
his letter to Lord Kelvin from Septem-
ber 7, 1898: ‘Hele-Shaw has some beau-
Figure 10. Hele-Shaw cell tiful photographs, very interesting to
you and me. By means of a thin stra-
tum of viscous liquid between close glass walls, flowing past an interruption in the
film, you can realise experimentally the theoretical stream lines in 2 dimensions in
a perfect fluid flowing round a body represented in section by the obstacle’ (see,
[96]).
Hele-Shaw published several papers on this method, the most known of which
is [42]. They are:
• Experiments on the flow of water. Trans. Liverpool Engn. Soc., 1897;
• Investigation of the nature of surface resistance of water and of stream line
motion under certain experimental conditions, Trans. Inst. Nav. Archit., 1898
[Gold Medal];
• Experimental investigation of the motion of a thin film of viscous fluid, Rep.
Brit. Assoc., 1898 [Appendix by G. Stokes]
• Experiments on the character of fluid motion, Trans. Liverpool Engn. Soc.,
1898;
• The flow of water, Nature 58 (1898), 33–36;
• Flow of water, Nature 59 (1899), 222-223;
• The motion of a perfect fluid, Not. Proc. Roy. Inst., 16 1899, 49–64.
To conclude this section let me mention that Universities of Bristol and Liver-
pool perpetuated Hele-Shaw’s contributions and established prizes in his memory:
• Hele-Shaw Prize (University of Bristol) to the students in their Final Year in
any Department with a good academic or social record not otherwise covered;
• Hele-Shaw Prize (University of Liverpool) for a candidate who has specially
distinguished himself in the Year 2 examination for the degree of Bachelor or
master of Engineering.
3. First steps
3.1. Mathematical treatment
The first written mathematical treatment of the phenomenon discovered by Hele-
Shaw appeared in the famous monograph by Lamb [62]. Sir Horace Lamb (1849–
1934) created one of the most famous texts in Fluid Dynamics known as ‘Hydro-
dynamics’ to most applied mathematicians. The first edition of this work appeared
From the Hele-Shaw Experiment to Integrable Systems 9
in 1879 [61] as ‘Treatise on the Motion of Fluids’, a time at which the subject ac-
tively developing, with intriguing experiments and new theories. When Lamb was
17 years old he won a scholarship to Queen’s College, Cambridge (to read classics)
but soon he turned to mathematics and entered Trinity College Cambridge.
He was taught by Stokes and Maxwell and
graduated in 1872 with the second rank in the
list of those students awarded the First Class
degree. In 1875 Lamb moved to Australia where
he was appointed as a chair in mathematics at
Adelaide. He remained there with his family
for 10 years before taking an appointment as
a chair at the Victoria University in England
(now the University of Manchester). In 1920 he
retired and returned back to Cambridge where
he was awarded a honorary Rayleigh lecture-
ship, specially created for him. He was elected
as Fellow of the Royal Society in 1884. A part
of his life was dedicated to improvement and Figure 11. Sir Horace Lamb
completion of his famous book which run six
editions from the first [61] 258 page text to the sixth 738 page editions [63] then
reprinted several times, e.g. 1945, 1993. Chronologically they were 1879 1-st, 1895
2-nd, 1906 3-rd, 1916 4-th, 1924 5-th, and 1932 6-th. One sees that the book
matured simultaneously with the subject.
x3
h
x1,2
0
Figure 12. Velocity field in the cross-section of the Hele-Shaw cell
Lamb’s mathematical model for the Hele-Shaw cell was based on the fact that
a Newtonian fluid sandwiched between two closely situated parallel plates moves
according to the parabolic distribution of the velocity vectors V = (V1 , V2 , V3 ) in10 Alexander Vasil’ev
the (x1 , x2 , x3 ) space (Figure 12). Suppose that the flow is parallel and slow, i.e.,
∂V
= 0, V3 = 0.
∂t
Let us introduce the following notations:
p– pressure, v – velocity field, z = (x1 + ix2 )–
phase variable, µ– viscosity, h– the gap between
plates. Averaging across the vertical direction,
the Navier-Stokes equations reduce to
h2
v =− ∇p,
12µ
which is referred (see, e.g., [33]) as the Hele-
Shaw equation. The continuity equation leads
immediately to the Laplace equation for the
pressure p. ‘Folklore’ opinion suggests that
Stokes knew the Hele-Shaw equation already
at the time of the experiments, but I found no Figure 13. Porous medium
written evidence of this.
3.2. Darcy’s law
The Hele-Shaw equation reminds us of Darcy’s law in higher dimensions for flow of
a viscous fluid through a porous medium, where the flow is studied in a macroscopic
scale although the microscopic behaviour is rather complex (see Figure 13).
Averaging the slow flow equations through
the complicated microscopic random geometry
leads to a very simple equation
k
V = − ∇p,
µ
where k is permeability and µ continues to
stand for viscosity. This equation was obtained
in 1855 [12] experimentally by a French sci-
entists and engineer Henry Philibert Gaspard
Darcy (1803–1858). At that time he became
sick and retired from his position of Chief Di-
rector for Water and Pavements, Paris, and re-
Figure 14. H. Ph. G. Darcy turned back to his hometown Dijon where he
conducted research on viscous fluid filtration
through sand, which led him to the law called after him. He died in 1858 soon
after his discovery.
However it took a long time to derive Darcy’s law from the Navier-Stokes
equations via homogenization. Hubbert [48] and Hall [36] were the first who tried to
derive Darcy’s law by integrating the Navier-Stokes equation over a representative
volume element. However, these works were in a mechanistic fashion and wereFrom the Hele-Shaw Experiment to Integrable Systems 11
not sufficiently rigorous. Following these first attempts many authors improved
them (e.g., Poreh, Elata, Whitaker, Ahmed, Sunada, etc.). We do not include all
references to these works in order to keep the reference list reasonable for such a
paper however, the reader can find them, e.g., in [74]. Perhaps the most rigorous
derivation appeared only in 1977 by Neuman [74]. Darcy’s law became one of the
most important tools in Hydrodynamics and Engineering possessing numerous
applications, e.g., in oil recovery.
The next step was made by the Soviet scientist Leonid Samuilovich (Leib
Shmulevich) Leibenzon (1879–1951). One of the most prominent Soviet mechanists
of that time, he was born in Kharkov (Ukraine) in the family of a doctor. In 1897 he
entered Moscow University where he studied under N. E. Joukowski’s supervision.
During a difficult time in Russia Leibenzon traveled a lot working in different
institutions (in Tbilisi, Tartu, Moscow). In 1925 he organized the first laboratory
of the oil industry in the USSR. One of the most important researches of Leibenzon
was dedicated to Gas Dynamics.
The theory of the motion of gases in
porous media was developed by Leibenzon in
1921-1922 and was published in a series of
papers [64] in 1923, eight years earlier than
Muskat [72], see later complete publications in
[65; 73]. Approximately at that time the finite-
source model for the Hele-Shaw cell was pro-
posed, see Figure 16.
4. Conformal mapping
The finite sink/source model was first proposed Figure 15. L. S. Leibenzon
because of its applicability to the problem of in-
jection moulding. Molten polymer is forced into
a mould of appropriate shape through a strate-
gically placed hole [87]. The step from the local
behaviour of flow to the large scale evolution
yields many interesting consequences, the most important of which is the Free
Boundary formulation of the problem. In the case of a pointwise source/sink the
Hele-Shaw equation is reduced to the Laplace equation −∆p = Qδ0 (z), where Q
is the strength and δ0 (z) is the Dirac measure, and we leave the world of classical
functions and turn to distributions. The free boundary problem is formulated as
follows.
−∆p = Qδ0 (z), z ∈ Ω(t), (4.1)
p = 0, (4.2)
z∈Γ(t)12 Alexander Vasil’ev
Figure 16. Hele-Shaw cell with finite sink/source
where Γ(t) = ∂Ω(t),
∂p
= −v n . (4.3)
∂n z∈Γ(t)
Q > 0 for injection, Q < 0 for suction. Here Ω is the phase domain occupied by the
viscous fluid. Formally we use a complex variable z = x1 + ix2 instead of (x1 , x2 ),
which later becomes natural for the conformal treatment of the problem.
4.1. Polubarinova-Kochina and Galin
Polubarinova-Kochina and Galin were
the first to reformulate the free boundary
problem for the pressure as a boundary-
value problem for a conformal map from
some canonical domain (the unit disk in
our case) to the phase domain Ω(t). Let
us mention that this approach the be-
came one of the most widely used in field
theory, where such a map is called dy-
namical variables.
Pelageya Yakovlevna Polubarinova-
Kochina (1899–1999) died two months
before her centenary. A conference in
Figure 17. P. Ya. Polubarinova- Oxford in her honor became the con-
Kochina ference in her memory. Polubarinova-
Kochina was one of the most prominent
modern Russian-Soviet mathematicians and mechanists, a member of the Acad-
emy of Sciences, a laureate of many academic and governmental distinctions, a
Hero of Socialistic Labour (the highest award in Soviet Union), and a very kind
person. It is impossible to overestimate Kochina’s influence in Mathematics and
Industry. Her published works span the period from 1924 to 1999.
Pelageya Polubarinova was born in Astrakhan, a city situated in the delta
of the Volga River,100 km from the Caspian Sea. Her father Yakov StepanovichFrom the Hele-Shaw Experiment to Integrable Systems 13
Polubarinov, an accountant, discovered Pelageya’s particular interest in science
and decided to go to St Petersburg where she graduated from Pokrovskii Women’s
Gymnasium. In 1918, after her father’s death, Pelageya Polubarinova accepted
a job at the Main Geophysical Laboratory to bring in enough money to allow
her to continue her education. She worked under the supervision of Aleksandr
Aleksandrovich Friedmann (1888–1925). In 1921 she obtained a degree in pure
mathematics. In 1921–23 she met Nikolai Yevgrafovich Kochin (1901–1944) who
graduated from the Leningrad State University. They married in 1925 and had two
Figure 18. N. Ye. Kochin and P. Ya. Polubarinova
daughters Ira and Nina. In 1934 she returned to a full time position being appointed
as professor at Leningrad University. In the following year her husband N.Ye.
Kochin was appointed to Moscow University and the family moved to Moscow. In
1939 Kochin became Head of the Mechanics Institute of the USSR Academy of
Sciences, and a memeber of the USSR Academy of Sciences. Pelageya worked at
the same institute.
During the II World War Polubarinova-
Kochina and her two daughters were evacuated
to Kazan in 1941 when Germans approached
Moscow. However, N. Ye. Kochin remained in
Moscow carrying out military research. In 1943
she returned back to Moscow but Kochin be-
came ill and died soon after. He had been in
the middle of lecture courses and his wife took
over the courses and completed their delivery.
His research was on meteorology, gas dynamics
and shock waves in compressible fluids.
In 1958 P.Ya. Polubarinova-Kochina was
elected a member of the USSR Academy of Sci-
ences, and moved to Novosibirsk to help build- Figure 19. L. A. Galin
ing the Siberian Branch of the Academy of Sci-
ences. For the next 12 years she worked in Novosibirsk where she was Director
at the Hydrodynamics Institute and also Head of the Department of Theoretical14 Alexander Vasil’ev
Mechanics at the University of Novosibirsk. In 1970 she returned to Moscow and
became the Director in the Mathematical Methods of Mechanics Section of the
USSR Academy of Sciences.
One of her major contributions is the complete solution of the problem of
water filtration from one reservoir to another through a rectangular dam. In this
work she established connections with the Riemann P-function, Hilbert problems
and Fuchsian equations.
Lev Alexandrovich Galin (1912–1981) was born in Bogorodsk (Gor’kii re-
gion), graduated from the Technology Institute of Light Industry in 1939 and
started to work at the Mechanics Institute led by N. Ye. Kochin. He became a
professor at the Moscow State University from 1956 and Correspondent Member
of the Soviet Academy of Sciences from 1953.
P. Ya. Polubarinova-Kochina [79; 80] and L. A. Galin [23] independently gave
a conformal formulation of the Hele-Shaw problem in 1945,
η y
S1 z = f (ζ, t)
Γ(t)
ξ x
0 1 0
U Ω(t)
Figure 20. Conformal mapping
Q
Re [f˙(ζ, t)ζf ′ (ζ, t)] = , |ζ| = 1, (4.4)
2π
where z = f (ζ, t) = a1 (t)ζ + . . . is the time-dependent conformal mapping from
the closed unit disk |ζ| ≤ 1 onto the fluid domain Ω(t), and holomorphic and
univalent in the unit disk with f (ζ, 0) = f0 (ζ), see Figure 20. Polubarinova and
Galin proposed the first non-trivial solution to the free boundary problem with
suction taking a polynomial ansatz f (ζ, t) = a1 (t)ζ + a2 (t)ζ 2 . This allowed them
to reduce the problem to a simple system of ODEs for the coefficients a1 and a2 ,
see Figure 21. This solution revealed an awkward phenomenon of cusp formation
in the case of suction. The reason for such behaviour is that generally the problem
is Hadamard ill-posed for a receding viscous fluid and well-posed for an advancing
one. The equation (4.4) for the function f now is known as the Polubarinova-Galin
equation.
4.2. Kufarev and the existence of a solution
Taking different ansatzs one can obtain many explicit solution (e.g., polynomial,
rational, logarithmic). However, the problem becomes much more complex in itsFrom the Hele-Shaw Experiment to Integrable Systems 15
7.5
5
2.5
0
-2.5
-5
-7.5
-7.5 -5 -2.5 0 2.5 5 7.5
Figure 21. Polubarinova and Galin’s cardioide
general formulation: given an initial domain, and therefore, an initial function find
the solution to the Polubarinova-Galin equation. The problem of existence is very
difficult even for an advancing fluid. The Polubarinova-Galin equation can be easily
reformulated in the unit disk making use of the Cauchy-Schwarz representation.
The corresponding equation is a first-order PDE
Z 2π
˙ ′ Q eiθ + ζ
f (ζ, t) = ζf (ζ, t) dθ, |ζ| < 1.
0 4π 2 |f ′ (eiθ , t)|2 eiθ − ζ
This equation is of Löwner type, i.e., it repre-
sents a subordinating evolution of domains, see
e.g., [81]. The Löwner equation is linear and
the solution obviously exists. In contrast, the
above equation contains feedback in the driving
integral term. Kufarev with his student Vino-
gradov were the first to addressed the problem
of existence [101] in 1948.
Pavel Parfenievich Kufarev (1909–1968)
was born in Tomsk on 18 March, 1909. His
life was always linked with the Tomsk State
University where he studied (1927–1932), was
appointed as a docent (1935), and finally as
a professor (1944). He got the State Honor in Figure 22. P. P. Kufarev
Sciences (1968) just before his death. His main
achievements are in the theory of Univalent Functions where he generalized in
several ways the famous Löwner parametric method. But the first works were in
Elasticity Theory and Mechanics.
Kufarev was greatly influenced by Fritz Noether (Erlangen 1884– Orel 1941),
the brother of Emmy Noether, and by Stefan Bergman (1895–1977), who im-
migrated from nazi Germany (under anti-Jewish repressions) to Tomsk (1934).16 Alexander Vasil’ev
Bergman fortunately moved to Paris in 1937. Noether’s life turned to be more
tragic. He was arrested during the Great Purge, and sentenced to a 25-year impris-
onment for being a ‘German spy’. While in prison, he was accused of ‘anti-Soviet
propaganda’, sentenced to death, and shot in the city of Orel (Orlovskii Zentral
concentration camp) in 1941.
The results of Vinogradov and Kufarev’s 1948 paper and Kufarev’s other
results in exact solutions [100; 58; 59; 60] had remained unknown until Vladimir
Gutlyanskiı̆ and Yuri Hohlov brought them to Western audience in early 90’s.
The result of existence was reproved in modern language as a particular case of
the nonlinear abstract Cauchy-Kovalevskaya Theorem in 1993 by Reissig and von
Wolfersdorf [85]. Kufarev gave many exact solutions: when the initial domain is a
strip or a half-plane; when the initial domain is a disk with a non-centered sink;
rational exact solutions; the case of several sinks/sources, etc. It is interesting that
the review in Mathematical Reviews (MR0097227, 20# 3697) on the now famous
1958 paper by Saffman and Taylor [91] says ”...the authors do not seem to be
aware of the fact that there exists a vast amount of literature concerning viscous
fluid flow into porous (homogeneous and non-homogeneous) media in Russian and
Romanian”, referring first of all to Kufarev’s works.
4.3. Saffman and Taylor fingering
Of course, one of the most important steps in the early treatment of the Hele-Shaw
experiment was the phenomenon of fingering described by Saffman and Taylor in
1958 [91]. It is the most common instability type occurring for a receding viscous
fluid (ill-posed problem). It never occurs for an advancing fluid. At the same time
an analytic travelling wave solution, known now as the Saffman-Taylor solution,
became an important example of an exact solution which exists for all time for
the ill-posed problem. The fingering phenomenon and its prediction is especially
Figure 23. Instability of an interface moving towards a more viscous fluid
important in industrial applications, e.g, oil recovery. During primary recovery,
the natural pressure of the reservoir, combined with pumping equipment, brings
oil to the surface. Primary recovery is the easiest and cheapest way to extract oil
from the ground. But this method of production typically produces only about 10
percent of a reservoir’s original oil in place reserve. Getting as much oil as possible
out of a reservoir has always been the industrys prime goal. Where pressure has
dipped, this often involves using water as a means of flushing the oil out, in the
secondary phase. However, due to lower viscosity water forms long fingers in the
sand-oil medium apparently reaching the pumping tube. This ends the secondaryFrom the Hele-Shaw Experiment to Integrable Systems 17
Figure 24. Growth of a single long finger
phase and generally results in the recovery of 20 to 40 percent of the original oil
in place. The tertiary phase may use thermal or chemical recovery.
Sir Geoffrey Ingram Taylor (1886, London–1975, Cambridge) is cited as one
of the greatest physical scientists of the 20th century. He was born in St. John’s
Wood, London, his father was an artist, and his mother, Margaret Boole, came
from a family of mathematicians (his aunt was Alicia Boole Stott and grandfather
George Boole). One can observe some interesting connections with other Boole
family members. Margaret’s grandfather Colonel Sir George Everest (1790–1866)
was a British surveyor, geographer and Surveyor-General of India from 1830 to
1843. After the Great Trigonometric Survey of India along the meridian arc, the
famous mount was named in his honor. His niece, Mary Everest, married a famous
mathematician George Boole (1815 –1864). Their 5-th daughter, Margaret’s sister,
Ethel Lilian married Wilfrid Michael Voynich (Michal Habdank-Wojnicz), 1865–
1930, a polish revolutionary and the author of the Voynich medieval manuscript.
Ethel Lilian became a writer and she is widely
known as author of The Gadfly, 1897. By the
way, other sisters entered the academic career
as well: Alicia was a Lecturer in Geometry,
Lucy became professor in Chemistry, and Mary
Ellen was a teacher.
In 1910 Taylor was elected to a Fellow-
ship at Trinity College, and the following year
he was appointed to a meteorology post, be-
coming Reader in Dynamical Meteorology. His
work on turbulence in the atmosphere led to
the publication of ”Turbulent motion in flu- Figure 25. Sir G. I. Taylor
ids,” which won him the Adams Prize in 1915.
During World War II Taylor again worked
on applications of his expertise to military problems such as the propaga-
tion of blast waves, studying both waves in air and underwater explosions.
These skills were put to the service of scientists at Los Alamos when Taylor was
sent to the United States as part of the British delegation to the Manhattan project
between 1944 and 1945. In 1944 he also received his knighthood and the Copley
Medal from the Royal Society. His final research paper was published in 1969,
when he was 83.18 Alexander Vasil’ev
Philip Geoffrey Saffman (born 1931) was a
student of George Keith Batchelor (1920-2000),
FRS–1957, an Australian scientist, a student of
Taylor, and a founder of the Journal of Fluid
Mechnics (1956). Saffman is a Theodore von
Kármán Professor at the California Institute
of Technology, USA.
The Saffman-Taylor exact solution in its
conformal formulation is as follows. The func-
Q
tion f (ζ, t) = 2πλ t − log ζ + 2(1 − λ) log(1 + ζ).
maps the unit disk U minus the slit (−1, 0] onto
Figure 26. P. G. Saffman
the phase domain Ω(t). The parameter λ, the
relative width of the finger, is freely defined in
0 < λ < 1.
Im z
π
Ω(t)
Re z
−π
Figure 27. Saffman-Taylor finger
But in experiments λ was found to be close to 1/2 except in some very special
cases (very slow flow, Saffman’s unsteady solution). Why is λ = 1/2 selected? This
is a famous selection problem which Saffman and Taylor posed in 1958. They also
proposed to use a small surface tension as a selection mechanism. Despite many
attempts both analytic and numeric, no solution has been commonly recognized
as final. Activities in this direction were especially intensive in the 1980s and were
summarized and completed in [51]. Basically, many authors exploited a reduction
to to a non-linear eigenvalue problem. Several groups in 1986-87 presented nu-
merical results on instability of all finger widths except λ = 1/2, in the limit of
vanishingly-small surface tension. Another survey [92] on the selection problem was
written by Saffman in 1986. One of the most cited (especially by physicists) paperFrom the Hele-Shaw Experiment to Integrable Systems 19
on the fingering phenomenon is the review by Bensimon, et al. [6]. Among numer-
ous works in this direction let me distinguish a couple of mathematical treatments
which are closer in spirit to my work. Following Saffman and Taylor’s proposal
on the selection mechanism, Tanveer and Xie proved [98] that there is no classical
steady finger solution for nonzero surface tension and λ ∈ (0, 1/2). This confutes
previous numerical simulations. They also established that if the classical solution
for the finger problem with small nonzero surface tension exists for some λ ∈ (0, 1),
then it is close to a certain Saffman-Taylor solution, satisfies an algebraic decay at
infinity, and belongs to some special space of analytic functions. There can be a
discrete number of λ ∈ [1/2, 1) for which the selection problem has a solution under
this driving mechanism. An interesting alternative approach by Mineev-Weinstein
[70] rejects surface tension as a selection mechanism and proposes finger forma-
tion because of a special dynamics of exterior analytic singularities in a class of
solutions suggested by Dawson and Mineev-Weinstein earlier in [69].
5. Modern period
The development of the Hele-Shaw problem can be rather easily seen before 1970.
Splashes of interest followed with non-active periods. The number of authors
was restricted, though there were many brilliant scientists among them. The last
decades of XX-th century were characterized by a real boom of interest from all
fields of natural sciences and engineering. It is just impossible to summarize all
recent achievements and challenges in the topic. So I restrict myself to only a par-
tial overview of results led finally to what is now known as the Laplacian Growth
problem, applications of conformal mapping and some potential theoretic remarks.
5.1. Richardson and conservative quantities
While the title ‘Modern period’ is rather vague
and questionable, it is commonly accepted that
the modern period of the developments in the
Hele-Shaw problem started from a 1972 seminal
paper [87] by Stanley Richardson (1943–2008)
who suddenly passed away March 12, 2008 af-
ter having suffered a heart attack. Dr. Stan
Richardson was born near Macclesfield in 1943.
He studied at Cambridge, completing the four
Figure 28. S. Richardson years of the Mathematical Tripos in three, be-
fore going on to do his Ph.D. After 6 years in
Cambridge he came to Edinburgh. He was a
lecturer and then, a reader in the School of Mathematics, University of Edinburgh
from 1971. He published 22 papers which represent a master work and which
established Stan as a world class figure in the field.20 Alexander Vasil’ev
Let us consider the problem with injection (Q > 0) and let the solution
(classical or strong) to the Polubarinova-Galin equation exist for t ∈ [0, t0 ). Since
the free boundary moves in the normal direction and the normal velocity on the
boundary never vanishes, we have Ω(t) ⊂ Ω(s) for 0 < t < s < t0 . Richardson [87]
introduced the harmonic moments
ZZ ZZ
Mn (t) = z n dσz = f n (ζ, t)|f ′ (ζ, t)|2 dσζ ,
Ω(t) U
where dσz and dσζ denote area elements in the z- and ζ- planes respectively. He
proved that
M0 (t) = M0 (0) + Qt,
Mn (t) = Mn (0), for n ≥ 1.
This result yields important consequences, first of all, from the point of view of
general theory of motion. If a contour evolution possesses a Hamiltonian interpre-
tation, then conservative quantities must be its first integrals, or Hamiltonians in
a hierarchy. The system possesses symmetries due to the Noether principle. This
clear connection with integrable systems was discovered almost 30 years later by
a strong group of physicists and mathematicians (Mineev-Weinstein, Krichever,
Wiegmann, et al.) to whom I shall refer later in this paper.
Another important reminiscence comes from potential theory. The inverse
problem in potential theory asks whether it is possible to determine a domain given
its harmonic moments. Generally this is not possible even for the simply connected
case. In other words, could two domains differ having the same exterior gravity
potential? Novikov’s theorem [76] assures that if two bounded domains are starlike
with respect to a common point and have the same exterior gravity potential, then
they coincide. The situation is different for Hele-Shaw evolution. Given an initial
simply connected domain with smooth boundary, constant moments Mn , n ≥ 1,
and the area changing linearly in time, the unique advancing Hele-Shaw evolution
is defined, see [87].
These two key results were developed and generalized in many ways by fol-
lowers later.
5.2. Strong and weak solutions
When speaking about a strong, or classical, solution of a differential equation one
generally means that all functions and boundaries appearing should be smooth
enough and that the equations involved should hold in a pointwise sense. For the
Hele-Shaw problem it is convenient to introduce the notion of a smooth family of
domains [99]. We call a family of domains {Ω(t)} smooth if the boundaries ∂Ω(t)
are smooth (C ∞ ) for each t, and the normal velocity vn continuously depends on
t at any point of ∂Ω(t).
Then a strong solution of the Hele-Shaw problem is defined to be a smooth
family {Ω(t) : 0 ≤ t < t0 } such that (4.1–4.3) hold in a pointwise sense (the
function p will be uniquely determined by Ω(t) and will be smooth up to ∂Ω(t)). IfFrom the Hele-Shaw Experiment to Integrable Systems 21
the domains Ω(t) are simply connected it is equivalent to require the Polubarinova-
Galin equation (4.4) to hold.
Given a domain Ω(0) with smooth boundary it is known that in the well-posed
case Q > 0 there exists a strong solution of (4.1–4.3) on some interval [0, t0 ). In
the ill-posed case Q < 0 such a statement is true only if ∂Ω(0) is analytic (see,
e.g., [99]).
An equivalent definition of a strong solution can be stated as follows. A
smooth family Ω(t), 0 ≤ t ≤ T , as above, is said to be a strong solution for the
Hele-Shaw problem if the equality
d
Z
h(z)dσz = Qh(0) (5.1)
dt
Ω(t)
holds for any function h which is harmonic in an open set containing Ω(t).
For the well-posed version (Q > 0) of the Hele-Shaw problem (without sur-
face tension) there is a good notion of weak solution. It is based on the Baiocchi
transform, replacing pressure p in (4.1) by
Z t
u(z, t) = p(z, τ ) dτ. (5.2)
0
This type of transformation, with time t replaced by the vertical coordinate y,
was used by Baiocchi in [5] to obtain a variational inequality formulation of the
so-called dam problem. For the Hele-Shaw problem, weak or variational inequal-
ity formulations (in somewhat different disguises) were obtained around 1980 by
Elliott, Janovský [18], Sakai [88; 89], and Gustafsson [29]. See also [19].
Let Ω(t) be a strong solution to the problem (4.1–4.3). The equality (5.1)
can be replaced by the inequality ≥ for any test function φ ∈ C ∞ (C), which is
subharmonic in Ω(t), i.e., ∆φ ≥ 0 in Ω(t). Integrating we obtain that
Z Z
φ(z)dσz − φ(z)dσz ≥ Q(t − s)φ(0), (5.3)
Ω(t) Ω(s)
whenever s < t and with φ as above. This motivates the following definition.
A one-parameter family of bounded domains Ω(t), 0 ≤ t < ∞, containing
Ω(0) is said to be a weak solution if the inequality
Z Z
φ(z)dσz − φ(z)dσz ≥ Qtφ(0) (5.4)
Ω(t) Ω(0)
is satisfied for any function φ, which is subharmonic and integrable in Ω(t).
We remark that if the strong solution exists for t ∈ [0, T ), then it coincides
with the weak one, both being unique in this case. The function u is related to p
in the strong formulation by the Baiocchi transform (5.2).
A weak solution is unique forward in time but can branch at any time in the
backward direction. This occurs when a simply connected domain Ω(T ) for some
0 < T < ∞ can be obtained as a result of a strong simply connected dynamics22 Alexander Vasil’ev
Ω(t), or else, as a result of a weak dynamics G(t), where G(t) for t < T is multiply
connected with some holes to be filled in as t → T − . In [32] it was shown that for
a simply connected initial domain the classical solution does not branch backward
in time (if it exists).
Weak solutions are more ‘physical’ in the sense that the strong solutions
break down when the topology changes, even though we can continue injecting. In
the suction case strong solution breaks down and the blow-up time t0 appears at
the moment of cusp formation. The situation is even more complicated than this,
though. Polynomial solutions can develop cusps of order (4n − 1)/2, where n is a
positive integer at some critical time t0 . The solutions then blow-up and do not
exist beyond t0 . The solutions that develop cusps of order (4n + 1)/2 can continue
to exist beyond t0 (see [46]). Moreover, if the initial function is a polynomial of
degree n ≥ 2, then cusp formation is guaranteed before the moving boundary
reaches the sink [43]. Non-polynomial solutions can produce other scenarios of
evolution of the free boundary where, for instance, the blow-up time occurs at the
moment when the free interface reaches the sink.
As for initial free boundary singularities, interesting results were obtained for
the Hele-Shaw injection problem in the presence of angles at the boundary [53; 90].
It turns our that obtuse angles smooth instantaneously whereas acute angles can
persist for a finite time.
In [33] one finds a large collection of results on the inherited geometry for well-
posed Hele-Shaw flows. In particular, we proved that fingers can never occur under
injection. Another result says that the evolution with a starlike initial domain
exists for all time as a strong solution.
5.3. Regularization
There are several proposals for regularization of the ill-posed problem. One of
them is the “kinetic undercooling regularization” [44], where the condition (4.2)
is replaced by
∂p
β + p = 0, on Γ(t), β > 0.
∂n
It was shown in [44], [86] that there exists a unique solution locally in time (even
a strong solution) in both the suction and injection cases in a simply connected
bounded domain Ω(t) with an analytic boundary. We remark that at the con-
ference on Hele-Shaw flows, held in Oxford in 1998, V. M. Entov suggested to
use a nonlinear version of this condition motivated by applications. Reissig and
Pleshchinskii discussed this model in [78] where local existence and uniqueness
were obtained and some numerical results were presented.
Another proposal is to introduce surface tension as a regularization mecha-
nism. The model with nonzero surface tension is obtained by modifying the bound-
ary condition for the pressure p to be the product of the mean curvature κ of the
boundary and surface tension γ > 0 (Laplace-Young condition). A similar bound-
ary condition appears in metallurgy in the description of the motion of phase
boundaries by capillarity and diffusion.From the Hele-Shaw Experiment to Integrable Systems 23
The problem of the existence of the solution in the non-zero-surface-tension
case is more difficult. J. Duchon and R. Robert [17] proved the local existence
in time for the weak solution for all γ. Recently, G. Prokert [82] obtained even
global existence in time and exponential decay of the solution near equilibrium for
bounded domains. The results are obtained in Sobolev spaces W 2,s with sufficiently
large s. We also refer the reader to the works by J. Escher and G. Simonett [20; 21]
who proved the local existence, uniqueness and regularity of classical solutions to
one- and two-phase Hele-Shaw problems with surface tension when the initial
domain has a smooth boundary. The global existence in the case of the phase
domain close to a disk was proved in [22]. In [21] the authors have given a rigorous
proof of the fact that homogeneous Hele-Shaw flows with positive surface tension
as a driving mechanism are volume-preserving and area-shrinking. Most of these
authors work with the weak formulation of the problem.
5.4. Quadrature domains
I place certain emphasis on this topic since Björn Gustafsson, a mathematician to
whom this volume and the present paper is dedicated, my friend, colleague and
co-author, has made a major contributions in it.
Björn was born in Stockholm in 1947 and
all his life is connected with this city and
with the Royal Institute of Technology (Kun-
liga Tekniska Högskolan, KTH), where he stud-
ied and obtained his doctors degree (TeknD in
Swedish) under the supervision of one of the
great mathematicians in Complex Analysis and
Potential Theory, Harold Shapiro. It is worth
mentioning that Björn’s choice of the institute
was caused by bad weather at the moment
of his decision. Nordic countries are especially
known for their unstable and unpredictable cli-
Figure 29. B. Gustafsson
mate. Björn’s choice was between Stockholm
University and KTH. At one moment he had
decided for Stockholm University and started going there by bike to sign up. How-
ever, after a few kilometers it started raining and Björn returned home. The next
day Björn had changed his mind and applied to KTH instead. This was the way
the decision was taken. Despite his great achievements, Björn is a modest and
humble person. His sincerity and cheerful good humor are appreciated by all who
meet him. A brilliant and careful mathematician, an excellent musician (violin),
and a great collaborator, Björn is talented both as a scientist and teacher. His
encouraging lectures and presentations always attract both mature scientists and
novices. During his prolific career he was interested in wide spectrum of mathe-
matical problems and his results always positioned himself among top specialists
in the field. In the present survey, I distinguish Björn’s works in Free Boundary24 Alexander Vasil’ev
Problems and in the closely related area of Quadrature Domains. In the preced-
ing section I already mentioned that he proposed and developed the notion of a
weak solution to Hele-Shaw problem (valid even in higher dimensions). Quadra-
ture Domains are intimately related to Hele-Shaw flows. Richardson’s paper [87]
first revealed these connections, and the two subjects have since then developed
largely in parallel. The initial start to intensive research of quadrature domains was
launched in the 1970s basically by the ideas and work of Shapiro and his collab-
orators (Aharonov, Sakai and, later, Putinar) and students (Gustafsson, Ullemar,
Karp, Shahgholian, Ebenfelt), see [28; 88; 95]. For numerous further developments
I refer to nice surveys written by Gustafsson, Putinar and Shapiro [31; 34; 95].
Basically, quadrature identity is an exact formula for the integral of a har-
monic or analytic function expressed as a sum of pointwise values of this function.
Let Ω be a bounded domain in the complex plane, and let there exist finitely many
points a1 , . . . , am ∈ Ω, and coefficients ckj ∈ C such that
Z Xm nXk −1
f dσz = ckj f (j) (ak ), (5.5)
Ω k=1 j=0
for all integrable analytic functions f in
PΩ. This is the quadrature identity and Ω
m
is a quadrature domain. The sum n = k=1 nk is called the order of the identity.
Aharonov, Shapiro and Solynin [1; 2; 3] discovered relations to certain extremal
problems for univalent mappings. Sakai [88] developed quadrature domains from a
potential theoretic point of view and Davis [15] discussed quadrature domains in
the context of numerical analysis and approximation. By the way, Davis introduced
the Schwarz function in 1958 together with Pollak [14]. Putinar [83; 84] discovered
beautiful connections between operator theory and quadrature domains based on
the exponential transform. While Caffarelli, Karp and Shahgholian [7; 50; 94] de-
veloped the general free boundary point of view on quadrature domains, Crowdy
[10] discussed applications of quadrature domains in Fluid Mechanics. The relation
with the Hele-Shaw problem follows from the definition of a weak solution (5.4),
where the equality is achieved
Z Z
h(z)dσz − h(z)dσz = Qth(0),
Ω(t) Ω(0)
for all h harmonic in a neighbourhood of the closure of Ω(t), Q > 0. It immediately
follows that if Ω(0) is a quadrature domain satisfying the identity (5.5), then the
evolution Ω(t) consists of quadrature domains for all t, with the origin as a new
quadrature point (if there was no such point at the starting moment).
A closely-related (to quadrature domains and to weak solutions) point of view
for Hele-Shaw flow was also developed by Gustafsson and Sakai and it is known
as partial balayage. Recently, Gardiner and Sjödin [24] have used partial balayage
to make major progress on classical inverse problems in potential theory. In the
Hele-Shaw context it means sweeping the Dirac measure out in evolution with the
prescribed density function (see, [30; 33]).From the Hele-Shaw Experiment to Integrable Systems 25
5.5. Integrable systems
The newest direction in the development of Hele-Shaw flow is related to Integrable
Systems and Mathematical Physics, as a particular case of a contour dynamics.
y
Z
• Mk = − z −k dσz ;
Ω+
−
• M0 = |ΩZ |;
Ω + • M−k = z k dσz ;
Ω−
x • k ≥ 1,
• t = M0 /π, tk = Mk /πk
0 generalized times.
Ω−
Figure 30. Interior and exterior domains
A list of complete references to corresponding works would be rather long so
we only list the names of some key contributors: Wiegmann, Mineev-Weinstein,
Zabrodin, Krichever, Kostov, Marshakov, Takebe, Teo et al., and some important
references: [55; 56; 57; 66; 71; 97; 105]. Following the definition of Richardson’s
moments let us define interior and exterior moments as in Figure 30. The inte-
grals for k = 1, 2 are assumed to be properly regularized. The main seminal idea
(see, e.g., [105]) of the above authors is that the moments satisfy the 2-D Toda
dispersionless lattice hierarchy
∂M−k ∂M−j ∂M−k ∂ M̄−j
= , = .
∂tj ∂tk ∂ t̄j ∂tk
The real-valued τ - function becomes the solution to the Hirota equation
∞
6 X 1 ∂ 2 log τ
Sf −1 (z) = 2 ,
z z n+k ∂tk ∂tn
k,n=1
where z = f (ζ) is the parametric map of the unit disk onto the exterior phase
domain and Sf −1 (z) denotes the Schwarzian derivative of the inverse to f .
C−k ∂ log τ C̄−k ∂ log τ
= , = , k ≥ 1.
π ∂tk π ∂ t̄k
The τ -function introduced by the ‘Kyoto School’ as a central element in the de-
scription of soliton equation hierarchies. The Polubarinova-Galin equation (4.4)26 Alexander Vasil’ev
can be rewritten as
∂f ∂ f¯
Q
Im = [u, v] = , θ = arg ζ,
∂t ∂θ 2π
with [·, ·] as Poisson brackets, f = u + iv. This equation is known as the string
constraint. The equation for the τ -function with a proper initial condition provided
by the string equation solves the inverse moments problem for small deformations
of a simply connected domain with analytic boundary. The connection extends to
the Lax-Sato approach to the dispersionless Toda hierarchy. In this setting it is
shown that a Laurent series for a univalent function that provides an invertible
conformal map of the exterior of the unit circle to the exterior of the domain, can
be identified with the Lax function. The τ -function appears to be a generating
function for the inverse map. The formalism allows one to associate a notion of
τ -function to the analytic curves.
In the paper [57], an analog of this theory for multiply-connected domains is
developed. The answers are formulated in terms of the so-called Schottky double
of the plane with holes. The Laurent basis used in the simple connected case
is replaced by the Krichever-Novikov basis. As a corollary, analogs of the Toda
hierarchy depend on standard times plus a finite set of additional variables. The
solution of the Dirichlet problem is written in terms of the τ -function of this
hierarchy. The relation to some matrix problems is briefly discussed.
As I mentioned already, Hele-Shaw evolution represents a special case of
subordination growth. The Löwner theory deals with general evolution. Let us
consider the Löwner evolution in two forms: the time-dependent function maps
the exterior of the unit disk onto a domain minus a time-dependent slit; the time-
dependent function maps the upper half-plane onto the upper half-plane minus
a time-dependent slit. The corresponding Löwner differential equation in partial
derivatives in the first case is called ‘radial’ and in the second case ‘chordal’. The
dispersionless Toda and KP hierarchies were described and were proved in [97] to
be connected with the radial and with the chordal case respectively. Their main
result for the radial Löwner equation reads as follows. A solution to the radial
Löwner equation represents the Lax function that solves the Lax equation for the
dToda hierarchy provided the dependence of the Lax on the generalized times is a
solution of a hydrodynamic-type system of equations. This system guarantees that
the dependence of the Lax function on generalized times is performed through a
single function, the solution to the system. The converse is also true. A one-variable
reduction of the hierarchy, i.e., the Lax function whose dependence on generalized
times is performed via a single function, satisfies the radial Löwner equation.
Analogously, the chordal Löwner equation is connected with dKP hierarchies. This
result is related to the eigenvalue distribution of normal random matrices in the
large N limit.
The above considerations lead to a Hamiltonian viewpoint of Hele-Shaw evo-
lution. Another point of view was proposed in [33; 104] and was based on the ideaYou can also read
