A NNALES DE L ’I. H. P., SECTION C
J OSEF B EMELMANS
On a free boundary problem for the stationary Navier-Stokes equations
Annales de l’I. H. P., section C, tome 4, n
o6 (1987), p. 517-547
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On
afree boundary problem
for the stationary
Navier-Stokes equations
Josef BEMELMANS Fachbereich Mathematik,
Universitat des Saarlandes, D-6600 Saarbrucken, Germany
Vol. 4, n° 6, 1987, p. 517-547. Analyse non linéaire
ABSTRACT. - We
investigate stationary
flows in a fluidbody together
with its free
boundary.
The fluid is assumed to be viscous andincompressi- ble ;
the freeboundary
isgoverned by continuity
of the normal stress.Hence the
configuration
to be considered here can beregarded
as ageneralization
of a classicalequilibrium figure.
The main tool inproving
existence of a
regular
solution consists in a hardimplicit
function theorem.Key words : 76D05, 76U05, 35 Q 10.
RESUME. - Nous etudions les flots stationnaires dans un corps fluide a frontiere libre. Le
liquide
estsuppose visqueux
etincompressible;
la fronti-ère libre est
regie
par la continuite de la contrainte normale. Lesconfigur-
ations que nous examinons ici peuvent etre
regardees
comme unegeneralis-
ation d’une
figure d’equilibre classique.
L’outilprincipal
pour demontre l’existence d’une solutionreguliere
est un theoreme de fonctionimplicite
ala Nash-Moser.
Annales de l’lnstitut Henri Poincaré - Analyse non linéaire - 0294-1449 Vol. 4/87/06/517/32/$5,10/(g) Gauthier-Villars
1. INTRODUCTION
Consider a
drop
of aviscous, incompressible
fluid under the influence of some exterior forcedensity f.
Astationary
flow inside thedrop
can bedescribed
by
the Navier-Stokessystem
together
with theboundary
conditionsAs
usual, v = (vl, v2, v3) = v (x),
denotes thevelocity, p = p (x)
the pressure, and v > 0 is the kinematicviscosity.
The unknown domainoccupied by
the fluid is calledQ,
itsboundary ~; n
is the outer normal toE,
and ti, t2 span thetangent plane.
With T
being
the stress tensor,D = (D1, D2, D3),
thedynamical boundary
conditions in(2)
state that the fluid cannot resist
tangential
stresses.Equation (3)
governs the freeboundary: E adjusts
itself such that thefluid’s
normal stressequals
.the
given
pressure po which is assumed to be constantthroughout
’
The
volume 1 Q of
thedrop
isprescribed,
too; after a suitable transforma- tion we canalways
obtain’
In this paper we prove the existence of classical solutions to the free
boundary problem (1)-(3)
for variousconfigurations.
Atypical
result iscontained in
THEOREM 1. - Let
Annales de l’lnstitut Henri Poincaré - Analyse non linéaire
be the
force of self-attraction.
Forf = fo + h,
with ~, > 6 and(in cylindrical
coordinates r,6, x3)
and
small
enough,
there exists aunique
solutionand
E E C6 + ~‘
to thefree boundary problem ( 1)-(3).
v and pare small in the sense that
and E lies in a
C6+ -neighborhood of
the unitsphere S;
theof
the distance
of 03A3 from
S canagain
be estimatedby C I
hRemark. - The solution established in this theorem can be
interpreted
as
perturbation
of a classicalequilibrium figure Eo
of arotating liquid.
Although Eo
is the trivialsolution, namely
the unitsphere,
whichprovides
an
equilibrium figure
for zeroangular velocity,
the method ofproof
isbased to a
large
extend on results for moregeneral equilibrium figures.
Therefore we can cover also other
geometrical configurations by
the sameapproach;
the details aregiven
inparagraph
6.To motivate our method of
proof
we compare theproblem (1)-(3)
withrelated free
boundary problems
for the Navier-Stokesequations.
Thetime-dependent analogue
to(1)-(3)
was solvedby
Solonnikov[14].
Heused the transformation
with
XeQ(O),
whereQ(t)
is the(unknown)
domainoccupied by
the fluid attime t;
V denotes thevelocity
as a function ofLagrangian variables,
i. e. the
velocity
of aparticle
at time t that wasinitially
atX E SZ (o). By (9)
the domain UQ(t)
can bemapped
onto thespace-time cylinder
orT
SZ (4) x (o, T)
which is a known domain since the initial datumQ(0)
isgiven.
As the unknown V is contained in(9)
the transformation leads toa
highly
nonlinear system;however,
the kinematicalboundary
conditionfor v. n is
automatically satisfied,
and hence the Navier-Stokesequations
with four conditions on the free
boundary
are reduced to a NeumannVol. 4, n° 6-1987.
problem
on agiven boundary.
Thiselegant
device(~)
cannot be used forthe
stationary problem (1)-(3)
because a flow is calledstationary
if v andp,
regarded
as functions of theposition
x rather than of theLagrange
variable X do not
depend
ontime;
as function of X astationary velocity
field does
change
in time except for trivial cases.Very
often the method to handle aparabolic problem
is modeled after the one used in theelliptic
case; here we meet
quite
a different situation.As a reduction to a
boundary
valueproblem
on a fixed domain seemsnot to be available we will
apply
successiveapproximations.
This method turned out to be useful in the related freeboundary problem
forstationary
viscous flows where surface tension governs the free
boundary, cf [1];
then
(3)
has to bereplaced by
where H denotes the mean curvature of
E,
and K= Const. is the coefficient of surface tension. To solve(1), (2), (10)
we construct a sequencef
v", p",~n ~n
1, where(vn, pn)
is the solution to theequations
of motion(1), (2)
in the domain that is boundedby En _ 1,
and1:n
can bedetermined from
( 10)
when T is evaluated at ToP E C1 +Ji
the mean curvature
equation ( 10) yields
a and on the otherhand,
for a solution v and p to(1), (2)
the normII
v~C2+ + ~p~C1+
canbe estimated
by
the of theboundary
E of theunderlying
domain. In this way the
approximation procedure
can be carried out inall
approximations {vn, pn, 03A3n},
if defined asabove,
lie in this space.
Successive
approximations
canyield only
local existencetheorems;
there-fore we can restrict ourselves to free boundaries E that lie in a
neighbor-
hood of a
given boundary ~o.
In thesimplest
case, whenEo
will be the unitsphere S,
E can berepresented
as agraph
over S:If we describe E in this way
(3)
becomes anequation
for the scalarunknown
ç.
( 1) Lagrangian variables are commonly used when the kinematics and dynamics of fluid
motions are studied in order to derive the equations of motions. Once these are established
one usually works with Eulerian variables. This seems due to the fact that almost all mathematical contributions about the Navier-Stokes equations concern problems where the
domain occupied by the fluid is fixed, and not free boundary problems.
Annales de l’lnstitut Henri Poincaré - Analyse non linéaire
If we
insert ç
into( 10)
it becomes asecond-order, elliptic equation
withdata of class hence the solution will be as is needed for the
approximation
scheme to converge.Formally, (3)
follows from(10) by letting
K tend to zero. As K is a coefficient in theprincipal part
of(10)
this indicates that
(3)
is no differentialequation
anymore for which ageneral
existencetheory
has beendeveloped.
To get a solution of(3),
theparticular physical background
must be taken into account, and it is obvious that forT E C1 + JI
thesolution §
will be at most of classC1 + JI,
too.
Compared
with theregularity
of(10)
above thisproperty
causes aloss of two derivatives in each
approximation step.
For suchproblems
hard
implicit
function theoremsprovide
a verypowerful
tool. This methodwas initiated
by
J. Nash[11]
and J. Moser[10];
here a version of Moser’s theorem due to Zehnder[18]
will beapplied,
it isparticularly
suited toour
problem
as will beexplained
inparagraph
2.We will close these
introductory
remarks with a brief discussion of the force of self-attractionfo
and of the local nature of our existence results.The free
boundary §
is determinedby
a first order differentialequation (3) which,
asjust remarked,
seems not to be solvable in thegeneral
case.Therefore we introduce the force
fo
ofself-attraction;
as it is agradient
its
potential
U can be absorbed into the pressure p in(2) ( 2),
and hence(3)
becomesNow the
unknown
appears in the domain ofintegration
Q as well as inn. If we
regard
the left hand side as theprincipal
part ofequation ( 12), then, according
to Lichtenstein[7],
theintegral
can bewritten in the form
where
03C80 (03BE)
is the normal derivative of the Newtonianpotential
ofS;
d (~, r~)
denotes the Euclidean distance between twopoints
The(2) A potential force in f does not affect v in (1), (2) since the pressure does not enter into the boundary condition (2).
Vol. 4, n° 6-1987.
nonlinear operator
N (~)
will be discussed inparagraphs 2, 4,
where we will also show that thesolvability
of( 12)
can be reduced to the existence of the inverse of theintegral
operatorM~.
In this way the introduction offo
asdominating
force leads to anequation
for the freeboundary
whichcan be handled. But also for
physical reasons fo
must beregarded
asnecessary. Self-attraction tends to hold the
drop together
and therefore balances other forces h that may act in theopposite
way.There exists a
counterexample
due toMcCready [9] which, although
found in
quite
a different context,supports
ourinterpretation
offo
asbeing
necessary.McCready
shows that it isimpossible
togive
an apriori
bound on Dirichlet’sintegral
of a solution v to the Navier-Stokesequations
if Neumann rather than Dirichlet conditions are
given.
Hence the basic apriori
estimate which was first provenby Leray
to establish the existenceof a
global
solution(i.
e. a solution forarbitrary large data f )
fails in thiscase. In the context of free
boundary problems
thisexample
indicates thatonly
local solutions exist because there are nolonger rigid
walls which hold the fluidtogether regardless
of thepossibly
verylarge
forces that generate the fluid’s motion.2. THE LINEARIZED
EQUATIONS
The free
boundary problem (1)-(3)
can beregarded
as an abstractequation
where we define F
by assigning
toz = (v, p, E)
theright-hand
sides of(1)-(3).
As we want to solve(14) by
means of a hardimplicit
functiontheorem we have to
investigate
the linearization of FI am indebted to Professor S. Hildebrandt for aquainting me with Lichtenstein’s paper
[7] and to Professor E. Zehnder who introduced me into the method of hard implicit function
theorems. This version of the paper was written at the Mittag-Leffler Institute in Djursholm.
It is a great pleasure to thank the institute and its director, Professor L. Hörmander, for their kind hospitality.
Annales de l’Institut Henri Poincaré - Analyse non linéaire
but this can
only
be defined if the domain of F admits an affine structure:if
(v, p)
isgiven
on fi whereaS2 =.E,
and if vand fi
are defined on theclosure of another
domain 03A9
with we have to define their sumX) + ( v, p, E),
at least in aneighborhood
of somegiven configur-
ation zo =
(vo, po,
As we indicated
already
in( 11)
we restrict ourselves to surfaces E thatare
graphs
over some known closed surfaceEo:
where no
(xo)
denotes the normal toEo
at apoint
xo EEo.
In certain cases we may use coordinates
(~1, ~2, p)
on(~3,
where~ = (~1, ~2)
are local coordinates on this leads to thesimpler expression
Here we may think of
spherical
orelliptic
coordinates. Due to the local structure of our existence theorems we canalways
assumeI ~ I 1;
thedomain included
by
X is then of the formand
similarly
in thegeneral
case. We now can define the « sum » of two boundaries E andE by
where § and ç
are the scalar functionsdescribing
E andE
resp. in(17).
Therepresentation
of £by ç
leadsimmediately
to a one-to-onetransformation of Q onto a standard domain
Qo (which
is the one boundedby Xo);
for this reason consider for(ç, p) eD
themapping
From our
assumption ] §
1 it is obvious that a isinvertible;
furthermorea as well as when considered as a function
of ç
and p are asregular as § (~);
derivatives D°‘ a, D°‘ 6-’with a ~
k can be estimatedby
derivatives
of ç
of the same order and vice versa. If weassign
topoints (03BE, p)
~03A9 and03C3(03BE, p) E S2o
its cartesian coordinates x and y, the transforma- tion(2)
definesunambiguously
an invertible relation between the cartesianVol. 4, n° 6-1987.
coordinates x~03A9 and
y~03A90
which we denoteagain by
a:Remark. - We note that it is also
possible
to choose a domainQ~
instead of
Qo
whichbelongs
to the solution zo of( 14),
such thatQ,
andall
quantities
defined onit,
arerepresented by
functions on HereQ~
isdefined
by
a 1-to-1mapping
If we call the transformation from Q toQ~ again
a, the structure of theequations (25)-(35)
below isprecisely
the same. As we will see inparagraph
3 anappropriate
choiceof
Q~
sometimesyields
a considerableadvantage.
Using
the transformations from(21)
we can now write thedependent
variables v and p as functions of
Y E Qo;
in this way it becomespossible
todefine the sum of two
velocity
fields v andv
that areinitially given
ondifferent domains Q and
Q.
We setagain
withx E SZ, Y E Qo
and y = a(x).
Remark. - The
advantage
of the transformation(22)
over thesimpler
one
u‘ (y):
= v‘(6-1 (y))
which oneprobably expects
consists in the fact that(22)
maps fieldsv (x)
into solenoidal fields uagain,
nowdivergence-free
with respect to the new coordinates:(y)
= 0. If on the other hand theregularity
of v and u isconsidered, (22)
seems to lead to a certaindisadvantage
since for the newfunction u is of class
C2 only
if a is three-times differentiable. As we willsee later in Lemma 4 we need three derivatives of £
(that
is ofa)
anywayto bound second derivatives of u if u solves the
equation
of motions.Hence, as far as solutions to
( 1), ( 2)
and their linearizations areconcerned, (22) only simplifies
theproblem.
Now let
(v, p)
and(v, p)
begiven
functions that are defined on Q andS2
resp., where the domains are characterizedby
scalarfunctions § and ç
as in
(18).
Let a and6
be the transformations(20)
that are inducedby ç
and
ç.
If we insert v, p, a andv, p, 6
into(22), ( 23)
we obtain astransformed
velocity
fields and pressures(u, q)
and(u, q).
These functionsnow have the same domain of definition
Q~
such thatM+M
andq + q
areAnnales de l’Institut Henri Poincaré - Analyse non linéaire
well-defined
quantities.
As we will see next ourfree-boundary problem
admits a formulation
in terms of the functions u :
Qo --+ 1R3,
q :go --~ f~, ~ : Eo -~
is thenpossible
to define the linerarizationD2 F (f, z).
If we
apply
the transformations(21)-(23)
to the Navier-Stokesequations ( 1)
we obtain after a formal but tedious calculationwith
Here
D~
meanspartial
differentiation with respect to the new variablesy’.
The various coefficients
depend
on the transformation a and its deriva-tives, namely
To indicate that L and the coefficients
depend
on o(and
therefore onQ
we sometimes write
L (~),
etc. It is understood that coefficients atJ,a, and (Xij etc.
(see below) depend
on a and its firstderivatives, ~t~
etc. on (7,
Do,
andfinally
Cij on o and its derivatives up to order three.Vol. 4, n° 6-1987. ’
In the same way the
boundary
conditions(2), (3)
aretransformed;
weobtain
’
As we outlined in
paragraph
1 aspecial
form of theequation (3)
for thefree
boundary
arises when the force of self-attraction10
is introduced.Instead of
(30)
we havewhere
Here we use the same
symbols
as in Lichtenstein’s paper[7],
where theintegral equation (31)
isderived; cf.
alsoFigure
below.Uo (y)
andU (y)
denote the Newtonianpotentials ~ y - y’ I ~ 1 dy’ over
Qo
and Q resp.; fory~E0
we writeU0(03BE)
instead ofUo(Y (03BE)).
Thedomain !7 is defined as
(S2BSZo)
U is the surface elementdoB
of
E~ --_ ~ yT = y + i~ (y) no (y),
formultiplied by
cos cp~, where cp~ is theangle
between the normals no andn~ ( y~) . Finally
d( ~, ~’)
is the Euclidean distance between
points ç and
of~o.
Annales de l’lnstitut Henri Poincaré - Analyse non linéaire
In this way we obtain the
following
formulation for( 14) : i = ~,..., 4 is of the form
together
with theboundary
conditionsF ~ ( f, z) = 0
becomeswith
M~ being
the linearintegral
operator as in(13).
F is then a
mapping
fromy0
x0
into whereVol. 4, n° 6-1987.
Next we compute the linearized operator
D2F ( f, z) z;
it is of thefollowing
form with
z = (u, q, ~) :
together
with theboundary
conditions for the operators in(40)
The
coefficients 1,
m, n are obtainedby
astraightforward
butlengthly
calculation. For the purpose of this paper,
however,
it suffices to concen-trate on their
regularity
in terms of theregularity
of u, qand ç
and theirsmallness if u, q
and ç
become small. Theseproperties
are listed below:(i)
is obtainedby differentiating Ni
i(u, Du, ~)
from
(36)
with respect to u. Thisgives, cf. (27), (28):
As a and
bijk
areanalytic in ç
their C0+ -norm is finite andsmall,
thecoefficients lij and
are small in the C° + ’-norm, too.(ii) l03B3 (u,
p,03B6)
denotes the coefficients of D03B303C3 where y is a multi-indexwith I y = 0,
...,3,
when we differentiateL (0
ui +Dj q
+Nt (u,
D u,0
with respect to o. As cr and its derivatives up to third order occur in these
expressions
we get thesum L
where we collect inl y l s 3
ly
the derivatives of all coefficients with respect to DY cr.Regularity
andsmallness of
the l03B3
are clear as theexpressions
in(28)
areanalytic
in aand its derivatives. This holds for all coefficients below in the same way.
Annales de l’Institut Henri Poincaré - Analyse non linéaire
(iii)
is derived fromdifferentiating
with respect to o.(iv) m o ( ~)
consists of derivatives of theintegral
inN ( ~)
from(34), (35)
with respect to
ç.
We obtain e. g.Note that U
depends
onç,
too. The smallness ofmo (~)
has beenproved by
Lichtenstein[7] (25).
(v) ry (u, q, ~)
are the coefficients that occur when G in(31), (33)
isdifferentiated with
respect
toç. As ç
and its derivatives up to second orderoccur we collect
again
the derivatives of the coefficients withrespect
toDY ç
and call them r~.(vi) m (~)
isgiven by differentiating
G withrespect
to uand q respectively.
In the
boundary
conditions(42)
wefinally
have(vii) ~}
fromdifferentiating
ai in(29)
with respect to~,
and-
(viii)
from 03B1ijk andTo
apply
theimplicit
function theorem we must invert the operatorD2 F ( f, z)
in aneighborhood
of zo. Butonly
at zo itself the system(40)-(42) splits
into a linearizedNavier-Stokes
system for( u, q)
and anequation
for~.
Forz ~ zo
all componentsof z
appear in eachequation (except
thefourth
one).
This establishes the essential difference to the
simpler approximation
scheme from
[1]
which was mentionedalready
inparagraph
1. There oneworks
only
with the"diagonal"
part of thegradient, namely DZi z),
i,j =1, ... , 4
andDz5
F~( f, z);
thisprocedure
however cannot be betterthan of first order and is therefore not
appropriate
for the hardimplicit
function theorem.
The
expressions Q
cannot beregarded
as smallperturbations
of the Navier-Stokes
equations
becausethey
contain third derivatives in theunknown a (that
is~).
Therefore the linearization admitsonly
anapproximate inverse;
before we can define it we have to recall Zehnder’simplicit
function theorem.Vol. 4, n° 6-1987.
3. ZEHNDER’S HARD IMPLICIT FUNCTION THEOREM AND THE APPROXIMATE INVERSE
The
implicit
function theorem on which this paper is based is formulated within thefollowing framework, cf.
Zehnder[18],
pp. 118-121.be a one-parameter
family
of Banach spaces withnorms I. It
such that forall t, t’
with0 _ t’ _ t
oo there holdsand
We assume the same
properties
to hold for{ ~Jt ~ t >_ o
and{ ~’t ~ t > o
also.As outlined
already
in(14), (39)
we aregiven
amapping
F from~ c
OY 0
withrange 9f
c such thatwith fo
and zo =(0, Uo (x), 0)
as in(6).
HYPOTHESIS H. 1. - Assume F :
~o -~ ~’o,
whereto be cont-inuous in
~( ~ z)
and two timesdifferentiable
in z;furthermore
there exists a constant
Mo
such .thatHYPOTHESIS H. 2. - For all
f z),
there holdsHYPOTHESIS H . 3. -
(F, fo, zo)
isof
order s with y as in H . 4 below. This meansAnnales de l’lnstitut Henri Poincaré - Analyse non linéaire
For every t E
[1, s]
there exists a constantMr
such thatfor
all( f, z) E (OJ/t
x n withHYPOTHESIS H . 4. - For every
( f,
there exists a linear mapH
( f, z) : ~’,~ -~ ~o
such thatH
( f, z)
isfurthermore
a continuousmapping from
intoprovided ( g z)
E~,~
(1 x Moreover there holds ..for
all( f, z)
thatsatisfy ( 52).
H
( f, z)
is an approximate inverse toD2
F( f, z)
in the sense thatholds
for
all cp EUnder these
hypotheses
we have thefollowing
hardimplicit
functiontheorem; .
and
‘~ :
~~, --~ ~P
is continuous whenever His.The numbers and p are chosen to be
p = 3, ~, > 6, cf.
alsoparagraph
6for a more detailed discussion of the minimal
regularity assumptions.
Vol. 4, n° 6-1987.
To prove the
hypothesis
above we chose thequantities
inthe
following
way:If
z)
isgiven
via a transformation of Q ontoQo [or Qt’ cf
the remarkabout the
mapping
a inparagraph
2(21)]
the coefficients areanalytic
functions
in f,
z,~, ~
etc. Hencecontinuity
in( f, z)
anddifferentiability
in z are
obvious; (47)
follows from the choice of the function spaces in(58).
For( f, implies
that the components of z, i. e. u, q,and
haveas many derivatives more as the order of the differential operators in F. Hence H. 1 is satisfied.
Lipschitz continuity
of F in the first argument followseasily
from(36):
which proves H. 2.
H. 3
again
isimplied by
theanalyticity of f and
the choice(58).
For(F, fo, zo) is
of infiniteorder,
i. e.(49)-(51)
is fulfilled for all s > 3.fo = DUo,
and
zo =(0, Uo, 0), cf [5]
areclearly C~-functions,
hence we have(49).
The inclusion
(50)
states that for moreregular
functionsf
and z insertedinto F the
image
is moreregular,
too. This is obvious from(36)-(38).
Thecoefficients
depend analytically
onf
and z, hence one can differentiate theseexpressions
with respect toyi
aslong
asf
and z allow it. The samereasoning applies
to(51).
To define the
approximate
inverseH(/z)
toD2 F (.I: z)
consider firstthe operator
D2
F(/ z)
that includes all ofD2 ( f, z)
in(40)-(42)
except theexpressions
that is consists of a linearized Navier-Stokes system in its first four components
solely
for the unknownu
andq,
and of anintegral
operatorfor ç
in its fifth component. We assume now, and this will be shown inparagraphs
4 and 5, thatD 2 F ( g z)
isinvertible,
and its inversewe denote
by
Then(53) holds,
for let begiven
withy = 3.
Then
cpt, i = 1,
..., 4, theright-hand
side in the linearized Navier-Stokes system is of classC 3 + ~,
hence(M, q)
xC4+J!, cf
Lemma4,
and(53)
Annales de l’lnstitut Henri Poincaré - Analyse non linéaire
holds
(~).
Thehigher regularity
of solutionsz = H ( f, z) ( cp)
ofDZ
F( f, z) (z)
= cp followsagain
from thetheory
of Navier-Stokesequations
and from the
integral equation in §, cf.
Lemmata 4 and 12. To estimate(D~
F( f,
H( f, z) -1) (cp)
we write thisexpression formally
aswhere
03C6’=(03C61, ...,03C64), 03C6"=03C65; D*2 F (f,z)
isthen (0
and b denote the
expressions
that have been deleted from Now consider ~B’ ~ (p"
andJA ’~(p~+~B’~ (p".
For (p 6~
we havez=(A-103C6’,B-103C6")~03B3,
therefore because of the diffe- rentiations in etc. Now theexpressions
in(59)
are small in thefollowing
sense;because §
occurs at most with its thirdderivative, cf (40)-(42).
This is notsufficient to prove
(55)
becauseif ç
gets close to the solution of the nonlinearproblem ~l03B3~
does not tend to zero asrequired.
At thispoint
we
exploit
thepossibility
ofchoosing
as reference domain another thanQo.
All estimates will beapplied
to asequence {zk}
whose convergence towards a solution of(14)
we show inparagraph
6. So when Zk+1 is constructed we can chooseQk-1
as reference domain(or
any other closeenough
toit).
Then the boundariesEk+ 1
andEk
arerepresented by
functions which we call
again ~k+ 1, ~k
via the relation(~1, ç2)
local coordinates onEk_ 1 ~. (61)
and all other
expressions
in(59)
are still estimatedby (60),
( 3) That we do not loose any derivatives here, as is allowed by H. 4 does not lead to any
improvement. The essential inequality is (54), where the loss of derivatives occurs.
Vol. 4, n° 6-1987.
but with the
special
choice in(61)
we obtainas
required by
H. 4.Remark. - This
procedure
should becompared
with the ones ofRabinowitz
[13]
and Kato[4],
where asingular perturbation problem
is studied for
periodic
functions and u. Thisproblem
is similar toours as the
perturbation f
containshigher
derivatives than theelliptic
operator that can be inverted. To obtain
approximate
solutions Rabinowitz introduces the
elliptic regularization
where
A~
is the linearisation off.
This deviceyields
even veryregular approximate
solutionsdepending
on the m one chooses. Acompletely
different
approach
is due to Kato[4],
where a new abstract existence theorem isgiven.
Thesingular perturbation problems
above arelocally
well
posed
if considered in Sobolev spaces, and therefore Kato proves existence without any loss of derivatives. It is not known whether this method can be carried over to the freeboundary problems
we are interestedin.
4. ESTIMATES FOR THE
LINEARIZED
EQUATIONS
OF MOTIONSThe
following
results forequations
of Stokes type are wellknown, cf Solonnikov-0160adilov [15],
Bemelmans[1].
LEMMA 3. -
(i)
Let begiven
whereQ’t is a
domain with
boundary of
classC3 + ~‘.
Then theboundary
valueproblem
Annales de l’lnstitut Henri Poincaré - Analyse non linéaire
with operators as in
(40), (42)
admits to a classical solutionas
long
are smallenough.
(ii) If SZ~
isrotationally symmetric
with respect to an axis~i,
the solutionu
isuniquely
determined modulo the rotation t[3
A y, t E R. For domainswithout this symmetry the solution is
always unique.
(iii)
The resultsof (i)
and(ii)
remain truefor non-homogeneous boundary data 03C8
aslong
asthey satisfy § n d a = 0.
LEMMA 4. - The solution
(u, q) of
Lemma 3 can be estimatedby
To extend Theorem 1 to the case where
~o
is notncessarily
the unitsphere
we have to solve theequations
of motion in a reference frame thatrotates with constant
angular velocity
mrelatively
to a fixed coordinate system; 03C9 is notgiven
but must be determinedby prescribing
the totaltorque exerted on the fluid
body.
If v denotes thevelocity
in the fixedframe,
u the one in the acceleratedsystem
then u and v are relatedby
A x. We then get the
following equations
ofmotion, cf Weinberger [17]
together
with the side conditionVol. 4, n° 6-1987.
In this formulation the coordinate system is chosen such that the torque
on Q vanishes.
If we denote
by ~’
the space of all C°°-functions on Q that aredivergence-free
andsatisfy
(p. Ax). n
onE,
we obtain aftermultiply- ing
theequations
of motionby
tp andintegrating
over QIntegration by
partsgives
where the surface
integral
vanishes due to(66);
furthermoreas the
integrals
and
all vanish. Because of div = 0 we have
If
J
is the closure of~’
with respectto II _ ( ~’ ( cp, cp) ) 1 ~ 2
we callv E J
a weak solution of
(65), (66)
if there holdsAnnales de l’lnstitut Henri Poincaré - Analyse non linéaire
for all For the linearized
equation
a weak solution is definedsimilarly.
The
equations (65), (66)
can be treated as in[1].
If 1: isrotationally symmetric
withrespect
to an axis co, then o A x isalways
atangential
vector and we have v. n = 0 as
boundary
conditions. Because ofwhere 2
(SZ)
is the closure ofC~03C3 (Q)
n{
v. n =0}
under K-normagain,
we can determine w
directly.
If E does not possess rotationalsymmetry
we where cpo are the pure rotations o A x, and the coerciveness of 5i
(v, v)
followsagi
an. Once a weak solution is established itsregularity
can be shown as for Lemma 4 above. Wefinally
remarkthat if we transform the
equations
as inparagraph 2,
the same existenceand
regularity
results hold. We formulate theseproperties
inLEMMA 5. -
Let f be in L2 (Q).
Then the linearizationof problem (65), (66)
admits aunique
solution in the classx. If
the data areregular
it canbe estimated as in
(64).
5. ESTIMATES FOR
LICHTENSTEIN’S INTEGRAL
EQUATION
The
integral equation
’
with the side condition that the volume included
by ç
isprescribed
can besolved for small data
~~
cpby
successiveapproximation, cf
e. g.Lichtenstein
[5],
pp. 14 ff.Again
we take someOT
as reference domain withboundary E~
of classC3 +~.
We now prove some estimates for solutions of(70)
we need later toapply
theimplicit
function theorem.Vol. 4, n° 6-1987.
LEMMA 6. - The
gradient of
thegravitational potential points inward,
i. e.
Proof. -
This property is well-known for classificalequilibrium figures, cf. [6].
ForE~
close to anequilibrium figure
E’ we haveaccording
to Lichtenstein[7] ( 11).
Here we denoteby
U’ the Newtonianpotential
of the domain boundedby 1:’,
n’ is the normal to E’ etc.(71)
enables us to normalize the operatorM;
it can be written in theform
with
We will
use Z
orrather §
whenever it is more convenient.LEMMA 7. - De
homogeneous integral equation
admits trivial
eigensolutions.
These are theinfinitesimal
translations~1, ~2,
~3
in the directionof
the coordinate axes and the rotations~4, ~s, ~6
aboutthese axes.
If Ez
isrotationally symmetric
about theyi-axis,
then~3
+is
noeigensolution,
i =1, 2,
3.Proof. -
Letbe the surface
E (which
isgiven by ~)
shifted into xl-directionby
theamount s, and let
QE
be the domain boundedby
Then the Newtonianpotentials of Q
and ofSZ£
are the same if evaluated atpoints
that differby
the same shift s ei :Annales de l’Institut Henri Poincaré - Analyse non linéaire
Therefore
according
to[7] (24).
As N(E~1) = o (E)
for E -0, cf [4] (25),
we obtainFor the rotations the
proof
followsalong
the same lines. IfE~
is rotation-ally symmetric
with respect to they3-axis
then and(n~ (y~ , yT ), n? (y~, y?))
areparallel
vectors on Hence~6
vanishes and is thereforeno
eigensolution.
LEMMA 8. - Consider the set M
consisting of
that partof
the branchof
MacLaurin
ellipsoids
that connects thesphere
with the branchof
Jacobiellipsoids.
For~~
E ~l theequation (73)
admits no othereigensolutions
thanthe ones obtained in Lemma 7.
Proof. -
All MacLaurin and Jacobiellipsoids
for which(73)
has inaddition to
~1, ... , ~6
from above othereigensolutions
have been charac- terizedby Ljapounoff [8], paragraphs
34 and 78. On M the first one with nontrivialeigensolutions
is theellipsoid corresponding
to the critical value of theangular velocity
at which the Jacobifamily
branches off.LEMMA 9. - Let
~T
be in JI. Thenand
admit the same solutions. Here
~2
denotes theintegral
operator with the iterated kernelhad
Vol. 4, n° 6-1987.