VOLUME-PRESERVING MAPS AND
GENERALIZED SOLUTIONS OF THE
EULER EQUATIONS
Yann Brenier
Abstrat.
The three-dimensional motion of an inompressible invisid uid is las-
sially desribed by the Euler equations, but an also be seen, following
Arnold [1℄, as a geodesi ona group of volume-preserving maps. Loalex-
isteneand uniqueness of minimalgeodesis have been established by Ebin
and Marsden [16℄. In the large, for a large lass of data, the existene of
minimal geodesis may fail, as shown by Shnirelman [26℄. For suh data,
we show thatthe limitsof approximatesolutions are solutionsof asuitable
extension of the Euler equations or, equivalently, as sharp measure-valued
solutionsto the Euler equations inthe sense of DiPerna and Majda [14℄.
1. Problems and results.
1.1. Lagrangian desription of inompressible uids.
Let D be the unit ube [0;1℄
d
in R d
(or the at torus T d
= R d
=Z d
), let
T >0beanitexed time and setQ=[0;T℄D. Theveloityeld of an
inompressible uid moving inside D is mathematially dened as a time
dependent, square integrable, divergene-free and parallel tothe boundary
Dvetoreldu. LetusdenotebyV thevetorspaeofallsuhvetoreds
having a suÆient smoothness, say ontinuous in (t;x) 2 Q and Lipshitz
UniversiteParis6etENS,DMI,45rued'ULM,75230ParisCedex, Frane.
ontinuous in x, uniformly in t. For eah u 2 V and eah a 2 D, we
an dene, aording tothe Cauhy-Lipshitz theorem, aunique trajetory
t2[0;T℄!g
u
(t;a)2D by
g
u
(0;a)=a;
t g
u
(t;a)=u(t;g
u (t;a)):
At eah time t, the map g
u
(t) = g
u
(t;:) belongs to G(D), the group (with
respet to the omposition rule) of all homeomorphisms h of D that are
Lebesgue measure-preserving inthe sense
Z
D
f(h(x))dx= Z
D
f(x)dx (1)
forallontinuous funtionsf onD. This set isinludedinS(D),the semi-
group of allBorel maps h of D that satisfy (1). We denote by G
V
(D) the
set fg
u
(T); u2V; T >0g and alltarget eah ofits elements h.
1.2. Minimal geodesis and the Euler equations.
Wedisuss the followingminimization problem : reah a given target h at
a given timeT >0with aeld u2V of minimalenergy
K(u)= 1
2 Z
Q
ju(t;x)j 2
dtdx (2)
(where we denote by j:j the Eulidean norm in R d
). In other words, nd a
minimizerfor
I(h)=inffK(u); u2V; g
u
(T)=hg: (3)
This amountsto look for a minimalgeodesis onnetingthe identity map
to h on the innite dimensional pseudo-Lie group G
V
(D) equipped with
the pseudo-Riemannian struture inherited fromthe Hilbert spae L 2
(D) d
.
This follows fromthe identity
K(u)= Z
T
0 Z
D 1
2 j
t g
u (t;a)j
2
dtda:
Wereferto[1℄,[16℄,[26℄,[2℄formoredetailsonthisgeometrialsetting. The
(formal) Euler-Lagrange equation of this minimization problem is nothing
but the Euler equation of inompressible uids (for whih we refer to [11℄,
[12℄, [19℄,[20℄, [22℄, et...), namely
r:u=0;
t
u+r:(uu)+rp=0 (4)
(where wedenotebya:bthe innerprodutoftwovetorsa,b inR d
,byab
their tensorprodut andbyr=(
1
;:::;
d
)the spatialpartial derivatives),
for some salar eld p = p(t;x), whih physially is the pressure eld. In
partiular, one an show that, if (u;p) is a smooth solution to the Euler
equation,then, for T >0 suÆiently smalland h=g
u
(T),I(h) isahieved
by u and only by u. (This is true as soon as T 2
<
2
, where is the
supremum on Q of the largest eigenvalue of the hessian matrix of p. See
[7℄, for instane, and Theorem 2..4 in the present paper.) So, the researh
of minimal geodesis is an alternative to the more lassial initial value
problem to study the Euler equations. (Of ourse, a better alternative
would beto look for allritial points of u!K(u) onV, when g
u
(T) and
T >0 are presribed, not onlythe minima.)
1.3. Non existene of minimal geodesis.
The lassial paperof Ebin and Marsden [16℄ shows that, if h belongs toa
suÆiently small neighborhood of the identity map for a suitable Sobolev
norm,thenthereisauniqueminimalgeodesionnetinghandtheidentity
map. However, inthelarge,a strikingresultof Shnirelman[26℄shows that,
for D = [0;1℄
3
and for a large lass of data, there is no minimal geodesi.
The argument is easy to explain. It is natural to denote by V
3
the vetor
spae V and V
2
the subspae of allplanar vetorelds, of the form
u(t;x)=(u
1 (t;x
1
;x
2 );u
2 (t;x
1
;x
2 );0):
Weonsiderdatah=g
u
(T)whereu2V
2
. Theyneessarilyare oftheform
h(x
1
;x
2
;x
3
)=(H(x
1
;x
2 );x
3
). We respetivelydenote by I
3
(h)=I(h) and
I
2
(h)the inmumofK onV
3 andV
2
. OfourseI
2
(h)I
3
(h). Assumethat
I
2
(h) > I
3
(h) (whih is possible [26℄,[28℄). Let us onsider a eld u 2 V
3
suh that K(u) < I
2
(h). The third omponent u
3
annot be identially
equal to0. Indeed, ifu
3
=0,then, foreveryxed x
3
, u(:;:;:;x
3
)belongsto
V
2
, and therefore
K(u)= Z
1
0
K(u(:;:;:;x
3 ))dx
3 I
2 (h):
Wean resaleu by setting (x
3
)=min(2x
3
;2 2x
3 ),
~ u
i
(t;x)=u
i (t;x
1
;x
2
;(x
3 ))
for i=1;2, and
~ u
3
(t;x)= 0
(x
3 )
1
u
3 (t;x
1
;x
2
;(x
3 )):
This new eld u~ still belongs to V
3
and reahes h at time T, but with an
energystritlysmallerthanK(u). So,I(h)anneverbeahieved,and,sine
the resaling proess an be iterated as often as we wish, we see that all
minimizing sequenes must generate small sales in the vertial diretion,
whih gives way to homogenization and weak onvergene tehniques to
desribe their limits.
1.4. Approximate solutions.
Let usintrodue the followingdenition of approximate solutions:
Denition 1..1 For >0, we say that a eld u
2V is an solution if
K(u
)+
1
2 jjg
u
(T)
hjj 2
L 2
(D) I
(h)+;
where
I
(h)=inffK(u)+ 1
2 jjg
u
(T) hjj 2
L 2
(D)
; u2Vg: (5)
Wealso dene
I(h)=lim
!0 I
(h)=sup
I
(h): (6)
From the denitions, we get
0I
(h)I(h)I(h)+1:
Of ourse, solutions dier from minimizing sequenes sine they don't
exatlymaththe nal onditions. Observe thateah minimizingsequene
isalsoasequene of
n
solutionsfor anappropriatesequene
n
!0,if and
only if I(h)= I(h). Aording to Shnirelman's papers [26℄,[28℄, this holds
true for all h 2 G
V ([0;1℄
d
) when d 3, but may fail if d = 2. (This is a
diret onsequene of the group property of G
V
(D) and of the key result
of Shnirelman showing that h ! I(h) is ontinous on G
V
(D) with respet
to the L 2
norm if and only if d 3.) Notie that the onept of solution
is less sensitive to the hoie of the vetor spae V than the onept of
minimizingsequenes, and makes sense when the target h only belongs to
the L 2
losureofG
V
(D),whihisknown tobethewholesemi-group S(D).
(See a detailedproof in[23℄.)
1.5. Measure-valued solutions to the Euler equations.
The existene of global weak solutions tothe initialvalue problem, for the
Eulerequations(4)indimension3,isunknown. (NotiehoweverthatLions
[19℄ has reently introdueda new onept of solutions,the so-alled dissi-
pative solutions, whih are globally dened for any initialveloity eld of
bounded energy,areweaklimitsoftheLeraysolutionstotheNavier-Stokes
equations when the visosity tends to 0and oinide with the unique loal
smoothsolutiontotheEulerequationswhentheinitialonditionissmooth.)
About ten years ago, DiPerna and Majda introdued an extended notion
of solutionsforthe Eulerequations,the soalled measure-valued solutions,
[14℄, based onaonept ofgeneralized funtions,introduedby Young [30℄
forthealulusofvariationsandusuallyalledYoung'smeasures,thatTar-
tarhadshown tobearelevanttoolintheeldofnonlinearevolutionPDEs
[29℄. DiPerna and Majda's measure-valued solutions an be desribed in
the followingway (up to the important energy onentration phenomenon,
widely disussed in [14℄ and here negleted). They are nonnegative mea-
sures (t;x;) of time, spae and veloity variables (t;x;)2 R
+
DR 3
satisfyingthe momentequations
Z
(t;x;d)=1; (7)
r
x :
Z
(t;x;d)=0; (8)
t Z
(t;x;d)+r
x :
Z
(t;x;d)+r
x
p(t;x)=0; (9)
for adistribution p(t;x),that an be interpreted asthe pressure eld. The
usual weak solutionsto the Euler equations orrespond to the speial ase
when is a Dira measure in . Just by using the natural bound on the
kineti energy (without any ontrol on the vortiity eld), DiPerna and
Majdaobtainedglobalmeasure-valuedsolutionsfromtheglobalweakLeray
solutionstothe3dimensionalNavier-Stokesequations,bylettinggotozero
thevisosityparameter. Unfortunately,(7),(8),(9)arenot aompleteset of
equationstoharaterizethesemeasure-valuedsolutions,sinethey involve
only their rst moments in . Letus mention a related onept of relaxed
solutions, due to Duhon and Robert [15℄, whih is more preise but does
not lead to any global existeneresult.
Our goalis toperformasimilar analysis,this time inavariationalontext,
for the researh of minimal geodesis when the existene of a (lassial)
solutionfails. Arstattemptwasmadein[4℄and,later,in[5℄,withapurely
LagrangianoneptofYoung'smeasures,theso-alledgeneralizedows. (A
generalized inompressibleow isdened as aBorel probability measure
on the produt spae = D [0;T℄
of all paths t 2 [0;T℄ ! !(t) 2 D, suh
that eah projetion
t
, for 0 t T is the Lebesgue measure on D and
the kineti energy, namely
Z
Z
T
0 1
2 j!
0
(t)j 2
dtd(!)
isnite. Wereentlydisoveredthatthis onepthadbeenalreadyused by
Shelukhin, inaslightlydierentframework[25℄.) Inthatsetting,weproved
the existeneofgeneralizedsolutions[4℄andboth existeneanduniqueness
of a pressure gradient, linked to them through a suitablePoisson equation
[5℄, but we failedin obtainingfor thema ompleteset ofequations beyond
the lassial Euler equations (see also [7℄). Suh an ahievement is made
possible in the present paper to desribe the limit of solutions as !0,
thanks to adierent (but related)onept of Young's measures, with both
Lagrangianand Eulerianfeatures, and,thanks toanappropriateregularity
resultonthepressuregradient. So,insomesense,weobtainsharpmeasure-
valued solutions tothe Euler equations.
1.6. The main result.
It is onvenient to make a lear distintion between the partile label and
their initialloation in D. The partile label is denoted by a and belongs
to ametri ompat spae A with a Borel probability measure da. For the
proofs of subsetion 3.8., the most onvenient hoie is A =T, where da is
theLebesguemeasure. ItisthenpossibletohooseiasaBorelisomorphism
transporting the Lebesgue measure from T to D. (Namely, i is one-to-one
up to anegligible set, and, for allontinuous funtion f on D,
Z
D
f(x)dx= Z
T
f(i(a))da:
Thisisalwayspossible,see[24℄,forinstane.) WedenotebyQ 0
theompat
metri spae QA, we onsider the spae of all Borel measures on Q 0
,
namely the dual spae of C(Q 0
) (the spae of all ontinuous funtion f on
Q 0
),with its(sequential)weak-*topology. Aordingtothe irumstanes,
we denoteby
< ;f >=<(t;x;a);f(t;x;a)>=
Z
Q 0
f(t;x;a)d(t;x;a)= Z
Q 0
fd ;
the pairingof a measure and aontinuous funtion f. We are now ready
tostate our main result:
Theorem 1..2 Assume that d = 3, D = [0;1℄
3
, Q = [0;T℄D and the
target h2S(D) satises
h(x
1
;x
2
;x
3
)=(H(x
1
;x
2 );x
3
); x2[0;1℄
3
; (10)
for some H 2S([0;1℄
2
). Then
i) I(h)3T 1
and K(u
)!I(h) for every solution u
as !0;
ii)Thereexistsauniquedistributionrp(t;x)intheinteriorof Q,depending
only on h, suh that, for every solution
t u
+(u
:r)u
+rp!0; (11)
in the senseof distributions in the interior of Q;
iii) rp is a loally bounded measure in the interior of Q;
iv) The respetively nonnegative and vetor-valued measures on Q 0
(t;x;a)=Æ(x g
u
(t;i(a))); m
(t;x;a)=u
(t;x)Æ(x g
u
(t;i(a))); (12)
have weak-* luster points (;m). For eah of them, the measure m is
absolutely ontinuous with respet to , with a vetor-valued density v 2
L 2
(Q 0
;d) d
, so that m=v, and the following equations are satised
Z
A
(t;x;da)=1; (13)
t +r
x
:m=0; (14)
t
(v)+r
x
:(vv)+r
x
p=0; (15)
in thesenseof distributionsin theinteriorof Q 0
, where isa naturalexten-
sionof , for whih the produt (t;x;a)r
x
p(t;x) iswell dened. Moreover
(0;x;a)=Æ(x i(a)); (T;x;a)=Æ(x h(i(a))) (16)
and the kineti energy
Z
DA 1
2
jv(t;x;a)j 2
(t;dx;da) (17)
is time independent, with value I(h)=T.
Remark 1.
Themeasure maynaturallybeonsidered asanelementofthe dualspae
of L 1
(dxdt;C(A)) (the vetor spae of all Lebesgue integrable funtion of
(t;x) 2 Q with values in the spae C(A) of all ontinuous funtion on A)
and
(t;x;a)=lim
Æ!0 Z
+1=2
1=2 d
Z
(t;x 2Æe Æy;a)(y)dy; (18)
holds true for the weak-* topology, when Æ ! 0, for any xed e 2 R d
and
any nonnegative smooth radial ompatly supported mollier on R d
. It
willbe shown that an be dened in asimilar way by
(t;x;a)=lim
Æ!0 Z
+1=2
1=2 d
Z
(t;x 2Æe Æy;a)(y)dy; (19)
for the weak-* topology of the dual spae of L 1
(jrpj;C(A)) (the vetor
spae of all jrpj integrable funtion of (t;x) 2 Q with values in the spae
C(A) of all ontinuous funtion on A). This shows that (t;x;a) oinide
with (t;x;a), as a probability measure (in a), for almost every (t;x) with
respettotheregularpartofjrpjandisitsnaturalextensiontothesingular
set of jrpj. Then, the produt rp makes sense. Notie that a similar
multipliation problemhas been suessfully handled by Majda and Zheng
[21℄ forthe Vlasov-Poissonsystem with singularinitialonditions (eletron
sheets).
Remark 2.
The limitmeasures (;m) are similartoYoung'smeasures [30℄, [29℄, witha
doubleharater, both Eulerianand Lagrangian,through their dependene
on both x (the Eulerian variable) and a (the Lagrangian variable). Equa-
tions (14), (15), (13) form a omplete set of equations for (;m). These
equationsarealreadyknowninthemathematialmodellingofquasineutral
plasmas [8℄,[17℄,[18℄ but the study of the orresponding initial value prob-
lemiswidelyopen, even forshorttimeandsmoothdata(see[18℄forpartial
results).
We may also onsider eah partile label a as a phase label, (:;:;a) and
v(:;:;a) as the orresponding onentration and veloity elds. Then, we
getamultiphaseowmodel,withaontinuumofphases,labelledbya2A.
For a disrete label a =1;:::;N, we would reover a lassial rude model
of N phase ows [10℄. This disrete model has been already studied from
a variational point of view in [6℄. As a matter of fat, some of the proofs
of [6℄an bestraightforwardlyextended to the ontinuous ase and willbe
reprodued(withfew hanges), for the seekof ompleteness, inthe present
paper(subsetions 3.2. and 3.3.).
Remark 3.
With eah solution (;m) of Theorem 1..2, we may assoiate a measure-
valued solution , inthe sense of DiPernaand Majda, by setting
Z
QR d
f(t;x;)d(t;x;)= Z
Q 0
f(t;x;v(t;x;a))d(t;x;a); (20)
for every ontinuous funtion f on QR d
, with at most quadrati growth
as ! 1 (sine v 2 L 2
(Q 0
;d) d
). Equations (7),(8),(9) are automatially
enfored, sine they are nothingbut (13),(14), (15), the two lastone being
averaged in a onA, with an obvious loss of information. So, the solutions
(;m)ofTheorem1..2maybeonsideredassharpmeasure-valuedsolutions
tothe Euler equations.
Remark 4.
Theuniqueness andthe partialregularityofthepressure gradientarestrik-
ingproperties, ertainlydue the variationalnatureof our problem. Forthe
initial value problem, there is no evidene at all that the pressure gradi-
ent should be aloallybounded measure in the large. The non uniqueness
of the luster points (;m) is not surprising, sine, already in the lassial
framework, I(h) may be ahieved by two dierent elds u. For instane,
for d = 2, D the unit disk, T = and h(x) = x, I(h) is ahieved by
both u(x
1
;x
2
) = ( x
2
;x
1
) and u. (Notie that the pressure is the same
for both, namely rp(x) = x.) It would be interesting to see whether or
not the uniqueness of the pressure gradient in the lassial framework an
be obtained by using only lassial arguments (without refering to weak
onvergene tools).
Remark 5.
Some families of (stationary) solutions (u
;p
) to the Euler equations be-
have in the same way as the solutionsdisussed in Theorem 1..2. Let us
onsider, for instane,
u
(x)=( os (x
1 )sin(
1
x
2
);sin(x
1 )os (
1
x
2 );0);
p
(x)=
1
4
(os (2x
1 )+
2
os (2 1
x
2 )):
As! 0,thepressure eld stronglyonverges(with bounded seondorder
derivatives) toward
p(x) = 1
4
os (2x
1 ):
Meanwhile, the veloity eld gets more and more osillatory and the mea-
sures(
;m
)denedby (12)onvergetoasolution(;m)ofequations(13),
(14), (15). A seondexample is
u
(x)=exp ( x
2
1 +x
2
2
2 )~u
(x);
~ u
(x)=(os ( 1
x
3
);sin(
1
x
3 );[ x
1 os (
1
x
3 )+x
2 sin(
1
x
3 )℄);
for whih the pressure is not even dependent
p(x)= 1
2
exp ( x 2
1 x
2
2 ):
2. Main steps of the proof.
2.1. Reformulation of the minimal geodesi problem.
The rst step is the omplete reformulation of our problemin terms of the
measures and m assoiated with u2V through
(t;x;a)=Æ(x g
u
(t;i(a))) (21)
and
m(t;x;a)=u(t;x)(t;x;a)=
t g
u
(t;i(a))Æ(x g
u
(t;i(a))): (22)
We automatially get (14). The initial and nal loations of the partiles
an beinluded in the weak formof this equationby
Z
Q 0
(
t
f(t;x;a)d(t;x;a)+r
x
f(t;x;a):dm(t;x;a)) (23)
= Z
A
(f(T;h(i(a));a) f(0;i(a);a))da;
foreveryf 2C(Q 0
)suhthatboth
t
f andr
x
f are ontinuous onQ 0
. The
inompressibility ondition an be expressed by
Z
Q 0
f(t;x)d(t;x;a)= Z
Q
f(t;x)dtdx; (24)
foreveryontinuousfuntionf onQ,that is R
A
(t;x;da)=1. Now,anele-
mentaryalulationshowsthatwean (abusively)rewriteK(u)=K(;m),
with
K(;m)= sup
(F;) Z
Q 0
(F(t;x;a)d(t;x;a)+(t;x;a):dm(t;x;a)); (25)
where the supremum is taken among all ontinuous funtions F and on
Q 0
, with values respetively inR and R d
, suhthat
F(t;x;a)+ 1
2
j(t;x;a)j 2
0; (26)
pointwise.
So, the minimizationproblembeomes
I(h)= inf
(;m) sup
(;p)
L(;m;;p); (27)
where the Lagrangian L isgiven by
L(;m;;p)=K(;m) Z
Q 0
[
t
(t;x;a)+p(t;x)℄d(t;x;a) (28)
Z
Q 0
r
x
(t;x;a):dm(t;x;a)
+ Z
A
((T;h(i(a));a) (0;i(a);a))da+ Z
Q
p(t;x)dtdx;
and measures , m are supposed tobeof form (21),(22).
2.2. Introdution of a relaxed problem.
If we relaxonditions (21),(22), we get a onvex minimization problemset
onthe spaeofmeasures(;m)(namelythe dualspaeofC(Q 0
)C(Q 0
) d
):
minimize K(;m) (dened by (25)) subjet to (23), (24). We denote by
I
(h)2[0;+1℄the inmum. Of ourse,I
(h)I(h). Then,we set :
Denition 2..1 An admissible solution is a pair of measures (;m) satis-
fying K(;m) <+1, (23) and (24). An optimal solution is an admissible
solution (;m) suh that K(;m) =I
(h).
ThenitenessofK(;m)<+1isenoughtoenforethatisanonnegative
measure positive on Q 0
, m is absolutely ontinuous with respet to , and
its(vetor-valued)densityv =v(t;x;a)issquareintegrableQ 0
withrespet
to :
m(t;x;a)=v(t;x;a)(t;x;a); (29)
K(;m)= Z
Q 0
1
2
jv(t;x;a)j 2
d(t;x;a): (30)
(This willbe shown in subsetion 3.2..) In subsetion 3.1., we prove :
Proposition 2..2 Let D=[0;1℄
d
andh2S([0;1℄
d
). Then, there isalways
an admissible solution (;m) and
I
(h)K(;m)dT 1
(31)
The study of the relaxed problem,in subsetions 3.2., 3.3., 3.4., 3.5., leads
to:
Theorem 2..3 Let D = [0;1℄
d
and h 2 S(D). Then there is at least one
optimal solution (;m) to the relaxed problem. We have m = v with v 2
L 2
(Q 0
;d) d
, equations (14), (13) and initial and nal onditions (16) are
satised, the kineti energy (17) is time independent and bounded by (31).
There existsa unique measure rp(t;x) inthe interior of Q, depending only
on h, for whih (15) holds true for all optimal solutions.
Then, in subsetion 3.6., we prove that the relaxed problem is onsistent
with the loalsmooth solutionsof the Euler equations.
Theorem 2..4 Let (u;p) be a smooth solution to the Euler equations (4),
T > 0 suh that T 2
<
2
, where is the supremum on Q of the largest
eigenvalue of the hessian matrix of p, and set h = g
u
(T). Then, the pair
(;m) assoiated with u through (21),(22) is the unique solution of the re-
laxed problem.
Next, in subsetion 3.7., we establish the following relationship between
solutions(asdenedinDenition1..1),andoptimalsolutionstotherelaxed
problem,through the assoiatedmeasures (
;m
) (dened by (12)) :
Proposition 2..5 Assume I(h)<+1. Let u
be an solution and (;m)
be any luster point(for the weak-* topology of measures onQ 0
) of (
;m
).
Then
I
(h)K(;m)liminfK(u
)limsupK(u
)I(h)=limI
(h): (32)
In addition, if I
(h)=I(h), then (;m) is optimal,
I
(h)=K(;m)=limK(u
)=I(h)=limI
(h) (33)
and (11) holdstrue.
Finally,in subsetion 3.8., we show :
Theorem 2..6 AssumethatD=[0;1℄
3
and h2S([0;1℄
3
)satises(10)for
some H 2S([0;1℄
2
), then I(h)=I
(h).
The proof is losely relatedto the one used by Shnirelman in [28℄ to show
thatlassialsmoothinompressibleowsaredenseamonggeneralizedows
(in the sense of [5℄), in the ase D =[0;1℄
3
. We believe that Shnirelman's
onstrutionan be adapted forany datah2 S([0;1℄
3
), but forthe sake of
simpliity and ompleteness, we only disuss in this paper the speial ase
(10), for whih we provide all details. This onludes the overview of the
main steps of the proof of Theorem 1..2.
2.3. Main steps of the study of the relaxed problem.
Insubsetion3.2.,elementaryargumentsfromlassialonvexanalysisshow
(asin [6℄) :
Proposition 2..7 Theinmum I
(h)isalwaysahieved andfor every>
0, thereexist someontinuousfuntions
(t;x;a) on Q 0
and p
(t;x) on Q,
with
t
, r
x
ontinuousonQ 0
and R
D p
(t;x)dx=0, suhthat, forevery
optimal solution (;m),
t
+
1
2 jr
x
j
2
+p
0 (34)
and
Z
Q 0
(j
t
+
1
2 jr
x
j
2
+p
j+jv r
x
j
2
)d 2
: (35)
Then, we get insubsetion 3.3. anapproximate regularityresult :
Proposition 2..8 Let 0< <T=2 and Q 0
=[=2;T =2℄DA. Let
x2D!w(x)2R d
be a smoothdivergene-free vetor eld,parallel toD,
and s 2 R ! e sw
(x) 2 D be the integral urve of w passing through x at
s=0. Then,
Z
Q 0
jr
x
(t;x;a) v(t;x;a)j 2
d(t;x;a)C 2
; (36)
Z
Q 0
jr
x
(t;x;a)j 2
d(t;x;a)C; (37)
Z
Q 0
jr
x
(t+;e Æw
(x);a) r
x
(t;x;a)j 2
d(t;x;a)( 2
+ 2
+Æ 2
)C; (38)
for all optimal solution (;m) and all , Æ and > 0 small enough, where
C depends only on D, T, and w.
Sine(t;x;a)is ameasure, possibly highlyonentrated (likea deltamea-
sure in x, as in the ase of lassial solutions), it is unlearhow todedue
fromProposition 2..8 a bound suhas
Z
Q 0
(j
t vj
2
+jr
x vj
2
)dC; (39)
by lettingrst !(togetv insteadofr
x
),then Æ; !0. Suh abound
wouldbemeaningful,ifouldbeboundedawayfromzero,whihisexatly
the ontraryof the lassialase and annot be expeted, anyway, beause
oftheinitialandnalonditions. Howeveraboundon R
jrpjisexpetable.
The formal(and, of ourse, not rigorous) argument is as follows. Starting
from(35), we get
(
t +
1
2 jr
x j
2
+p)=0:
Dierentiating inx, we formallyget (15) or
(
t
v+(v:r
x
)v+r
x
p)=0:
Then, integrating ina 2A,
Z
A (
t
v+(v:r
x
)v)(t;x;da)= r
x p;
and, by Shwarz inequality,
( Z
jr
x pj)
2
Z
j
t vj
2
d Z
d+ Z
jr
x vj
2
d Z
jvj 2
d:
All these alulations are inorret. However, the formalidea an be made
rigorous, by working only on the
and using nite dierenes instead of
derivatives, and lead (in subsetion 3.4.) to
Theorem 2..9 The family (rp
) onverges in the sense of distributions
toward a unique limit rp, depending only on h, whih is a loally bounded
measure in the interior of Q and is uniquely dened by
rp(t;x)=
t Z
v(t;x;a)(t;x;da) r
x :
Z
(vv)(t;x;a)(t;x;da); (40)
for ALL optimal solution (;m =v).
Finally,in subsetion 3.5., (15) is established.
3. Detailed proofs.
3.1. Existene of admissible solutions.
In this subsetion we prove Proposition 2..2, with an expliit onstrution
loselyrelatedtothe oneintroduedin[4℄forgeneralizedowsonthetorus
T d
(and used in [23℄ and [28℄). We only perform the onstrution in the
ases D=T d
andD=[0;1℄
d
, theseondone being anextensionofthe rst
one. (Extensions togeneral domainsan be performed as in[23℄.)
Admissible solutions on the torus.
Let h2 S(D) where D=T d
. We introdue, for (x;y;z)2 D 3
, a urve t 2
[0;T℄ ! !(t;x;y;z) 2 D, made, for 0 t T=2, of a shortest path (with
onstant speed) going from x to y on the torus T d
, and, for T=2 t T,
of a shortest path (with onstant speed) going from y to z. Suh a urve
is uniquely dened for Lebesgue almostevery (x;y;z)2D 3
. Then, we set,
for every ontinuous funtion f on Q 0
=[0;T℄DA,
<;f >=
Z
Q 0
f(t;!(t;i(a);y;h(i(a)));a)dtdyda
<m;f >=
Z
Q 0
t
!(t;i(a);y;h(i(a)))f(t;!(t;i(a);y;h(i(a)));a)dtdyda:
(Intuitively,thisamountstodeneageneralizedowforwhiheahpartile
a is rst uniformlysattered on the torus at time T=2 and then fousedto
the target h(i(a)) at time T.) This makes (;m) an admissible solution.
Let us hek, for instane, (13), by onsidering a ontinuous funtion f
that does not depend on a and showing that < ;f >=
R
Q
f. We split
< ;f >= I
1 +I
2
aording to t T=2 or not. For t T=2, we have, by
denition, !(t;i(a);y;h(i(a))) = !(t;y;y;h(i(a))). Sine we work on the
torus T d
, !(t;y;y;h(i(a)))=y+!(t;0;0;h(i(a)) y). Thus
I
2
= Z
[T=2;T℄DD
f(t;y+!(t;0;0;x y))dxdydt
(sine a2A!x=h(i(a))2D is Lebesgue measure preserving)
= Z
[T=2;T℄DD
f(t;y+!(t;0;0;x))dxdydt= Z
[T=2;T℄DD
f(t;y)dxdydt
= Z
[T=2;T℄D
f(t;y)dydt
(by using twie the translation invariane of the Lebesgue measure on the
torus) and, doing the same for I
1
, we onlude that < ;f >=
R
Q
f. We
also get, for K(;m) the following estimate that does not depend on the
hoie of h2S(D)
2K(;m)= Z
[0;T℄DA j
t
!(t;i(a);y;h(i(a)))j 2
dadydt
=T=2 Z
DA ((d
D
(i(a);y)=(T=2)) 2
+(d
D
(y;h(i(a)))=(T=2)) 2
)dady
= 4
T Z
DD d
D (x;y)
2
dxdyd=T
(where d
D
(:;:)denotes the geodesi distane on the torus).
Admissible solutions on the unit ube.
Let usnow liftthe unitube tothe torus by shrinkingit by a fator 2 and
reeting it 2 d
times through eah fae of its boundary. To do that, we
introdue the Lipshitz ontinuous mapping
(x)=2(min(x
1
;1 x
1
);:::;min(x
d
;1 x
d
)); (41)
from [0;1℄
d
onto [0;1℄
d
, and its 2 d
reiproal maps, eah of them being
denoted by 1
k
, with k 2f0;1g d
, and mappingbak [0;1℄
d
, one-to-one,to
theube2 1
(k+[0;1℄
d
). Wealsodenethe(almosteverywhere) one-to-one
Lebesgue measure-preserving map
1
k
(a)=i 1
( 1
k
(i(a))) (42)
fromT intoitself. Given h2S([0;1℄
d
), we assoiate
~
h2S(T d
) by setting
~
h(x)= 1
k
(h((x));
when x 2 1
2
(k+[0;1℄
d
), k 2 f0;1g. We onsider the admissiblepair (~;m)~
assoiated with
~
h and onstruted exatly as in the previous subsetion.
Then we set
(t;x;a)= 1
2 d
X
k2f0;1g d
(t;
1
k (x);
1
k (a));
v(t;x;a)= 1
2 d
X
k2f0;1g d
[((v(t;
1
k (x);
1
k
(a))):r
x )℄(
1
k (x)):
Expliit alulations show that (;m) is an admissible solution for h and
that (asin [28℄)
K(;m) 2d
3T dT
1
:
Notie that it is important to use a ontinuous transform (so that the
partiletrajetories are properlyreeted atthe boundary ofthe unitube
and not broken) inordertokeep (23), but thereisnoneed touse aone-to-
one transform, whih makes the onstrution very easy.
3.2. Duality and optimal solutions.
In this subsetion, we prove Proposition2..7 by using the same elementary
arguments of onvex analysis as in [6℄. Let us x an admissible solution
(;m) (suh a solution exists, as just shown in subsetion 3.1., introdue
the Banah spae E =C(Q 0
)C(Q 0
) d
,and the following onvex funtions
onE with values in℄ 1;+1℄,
(F;) =0 if F + 1
2 jj
2
0; (43)
and +1 otherwise, for(F;)2 E,
(F;) =<;F >+<m;> (44)
if thereexistp2C(Q)and 2C(Q 0
)with
t
2C(Q 0
)and r
x
2C(Q 0
) d
suh that
F(t;x;a)+
t
(t;x;a)+p(t;x)=0; (t;x;a)+r
x
(t;x;a)=0 (45)
for all (t;x;a)2 Q 0
, and (F;) =+1 otherwise. The Legendre-Fenhel-
Moreau transforms (see [9℄, h. I, for instane) of and are respetively
given by
(;m)=sup f<;F >+<m;>; F + 1
2 jj
2
0g (46)
and
(;m)=sup < ;F >+<m m;> (47)
where (F;) 2 E satises (45). In (46), we reognize denition (25) of
K(;m). So K preisely is the Legendre transform of . We see that
K(;m)isniteifandonlyifisanonnegativemeasureandmisabsolutely
ontinuous with respet to with a vetor-valued density v 2 L 2
(Q;d) d
and, then,
K(;m)= 1
2 Z
Q 0
jvj 2
d:
Observe that
(;m)=0if (;m) satises
< ;
t
+p>+<m m;r
x
>= 0; (48)
for all suitable (;p), and
(;m) = +1 otherwise. Thus, sine (48)
exatly means that (;m) satises admissibility onditions (13) and (23),
the inmum of
(;m)+
(;m) is nothing but the inmum of K(;m)
among alladmissiblepairs (;m), namely I
(h).
Funtions and are onvex with values in ℄ 1;+1℄. Moreover, there
is atleast one point(F;B)2E, namely
F = 1; =0;
for whih is ontinuous (for the sup norm) and is nite. Thus, by
the Fenhel-Rokafellar duality Theorem (an avatar of the Hahn-Banah
Theorem, see [9℄, h. I, for instane), we have the dualityrelation
inff
(;m)+
(;m); (;m)2E 0
g (49)
=sup f ( F; ) (F;); (F;)2 Eg
and the inmum is ahieved. More onretely, we get
I
(h)=sup <;
t
+p>+<m;r
x
> (50)
with
t +
1
2 jr
x j
2
+p0:
Letustakeanyoftheoptimalsolutionsasareferenesolution(;m). Then,
the duality relationexatlymeans that, for all>0,there existp
and
,
suh that
t
+
1
2 jr
x
j
2
+p
0
and
1
2
<;jvj 2
><;
t
+p
>+<m ;r
x
>+ 2
:
Thus,
1
2
<;jv r
x
j
2
>+<;j
t
+p
+
1
2 jr
x
j
2
j>
2
:
So, we have obtained the desired relationship (35) between the p
and the
andany any ofthe optimalsolutions(;m) andthe proofofProposition
2..7 is now omplete, notiing that R
D p
(t;x)dx = 0 an be enfored by
adding tothe
(t;x;a) a suitablefuntion of t.
3.3. An approximate regularity result.
Inthis subsetion again,weuse someargumentsof [6℄toproveProposition
2..8. Let (;m = v) be an optimal solution. Let Æ and be two small
parameters and 0 < < T=2. Let be a smooth ompatly supported
funtion on℄0;T[. Letx!w(x)be asmooth vetoreld on D,parallelto
D, and let e sw
(x) be the integral urve passing through x at s = 0. We
set
(t;x;a)=(t+(t);x;a); v
(t;x;a)=v(t+(t);x;a)(1+ 0
(t));
and, for allf 2C(Q 0
),
Z
Q 0
f(t;x;a)d
;Æ
(t;x;a)= Z
Q 0
f(t;e Æ(t)w
(x);a)d
(t;x;a); (51)
Z
Q 0
f(t;x;a)dm
;Æ
(t;x;a) (52)
= Z
Q 0
f(t;e Æ(t)w
(x);a)(
t +v
(t;x;a):r
x )e
Æ(t)w
(x)d
(t;x;a):
Notie that the new pair of measures (
;Æ
;m
;Æ
=
;Æ
v
;Æ
) is well dened,
has nite energy and satises (23). (Use the hain rule and ompute
Z
Q 0
d
dt [f(t;e
Æ(t)w
(x);a)℄d
(t;x;a):)
However, (13) isnot satised unless w is divergene-free. Letus show:
Proposition 3..1 Let (;m) be an optimal solution. Then, for any ;Æ,
Z
p
d
;Æ Z
p
d+
1
2 Z
j(
t +v
:r
x )e
Æw
r
x
Æe
Æw
j 2
d
(53)
2
+ 1
2 Z
j(
t +v
:r
x )e
Æw
j 2
d
1
2 Z
jvj 2
d:
Proof.
Usingdenitions (51), (52), we get the following identities,
Z
t
(d
;Æ
d)= Z
r
x
:(dm
;Æ
dm)
= Z
((
t +v
:r
x )e
Æw
):r
x
Æe
Æw
d
+ Z
v:r
x
d
= 1
2 Z
j(
t +v
:r
x )e
Æw
r
x
Æe
Æw
j 2
d
1
2 Z
j(
t +v
:r
x )e
Æw
j 2
d
1
2 Z
jr
x
Æe
Æw
j 2
d
+ Z
v:r
x
d:
Wealso have
Z
jr
x
j
2
(d
;Æ
d)= Z
jr
x
Æe
Æw
j 2
d
Z
jr
x
j
2
d:
Thus,
Z
(
t
+
1
2 jr
x
j
2
)(d
;Æ
d)
= 1
2 Z
j(
t +v
:r
x )e
Æw
r
x
Æe
Æw
j 2
d
1
2 Z
j(
t +v
:r
x )e
Æw
j 2
d
1
2 Z
jr
x
vj
2
d+ 1
2 Z
jvj 2
d:
Sine (;m) is anoptimal solution,we dedue from Proposition2..7
2
Z
(p
+
t
+
1
2 jr
x
j
2
)(d
;Æ
d)
+ 1
2 Z
jr
x
vj
2
d;
whihompletes the proof of (53).
Now, weuseProposition3..1withdierenthoiesofparameters;Æ. First,
in(53),weset Æ=0(sothate Æ(t)w
(x)=x). Therst termoftheleft-hand
side vanishes(sine R
A
(t;x;da)=1and R
D p
(t;x)dx=0). Thus, we get
1
2 Z
jv
r
x
j
2
d
(54)
2
+ 1
2 Z
jv
j 2
d
1
2 Z
jvj 2
d:
Wean bound frombelow the left-hand side by zero, let !0and get
0 Z
jv
j 2
d
Z
jvj 2
d (55)
foranyhoieof and smallenough. Thisimpliesthatthekineti energy
dened asa measure on[0;T℄by (17) is time independent (therefore equal
to I
(h)T 1
) and the right-hand side of (55) is bounded by C 2
, where C
depends onlyon D, T and . Thus, wehave obtained
Z
jv
r
x
j
2
d
( 2
+ 2
)C: (56)
Next, we assume w to be divergene-free, whih makes e sw
a Lebesgue
measure-preserving map for all s 2 R and anels the rst term of the
left-hand side in(53). So wean rewrite
Z
j(
t
+v:r
x )e
Æw
r
x
Æe
Æw
j 2
d (57)
2
2
+ Z
j(
t
+v:r
x )e
Æw
j 2
d Z
jvj 2
d:
Let us bound from below the left-hand side by zero and set = = 0 in
the right-hand side. We get
0 Z
(j(
t
+v:r
x )e
Æ(t)w
(x)j 2
jvj 2
)d:
Sine e Æ(t)w
(x)=x+Æ(t)w(x)+0(Æ 2
), this impliesthat
Z
v:(
t
+v:r
x
)(w)d=0 (58)
for all smooth ompatly supported in ℄0;T[ and all smooth divergene-
free parallelto D vetor eld w (whihmeans that
t Z
A
v(t;x;a)(t;x;da)+r
x :
Z
A
v(t;x;a)v(t;x;a)(t;x;da)
is agradient in the sense of distribution). Wean nowrewrite (57)
Z
j(
t
+v:r
x )(e
Æw
(x) x)+v(t;x;a) r
x
(x;e
Æw
(x);a)j 2
d (59)
2
2
+ Z
j(
t
+v:r
x )(e
Æw
(x) x)+vj 2
d Z
jvj 2
d
and, after rearrangingthe squares,
1
2 Z
jv r
x
Æe
Æw
j 2
d
2 Z
[v r
x
Æe
Æw
℄:[(
t
+v:r
x )(e
Æw
(x) x)℄d
+ Z
v:(
t
+v:r
x )(e
Æw
(x) x)d:
Sine e Æ(t)
(x) x=Æ(t)w(x)+0(Æ 2
)wededue from(58) that
Z
jv r
x
Æe
Æw
j 2
dC(
2
+Æ 2
); (60)
where C depends only onD, T, and w.
Finally, to get (38), assume that takes its values in [0;1℄ and satises
(t) = 1 if t T and (t) = 0 if t =2 or t T =2. For
(t;x;a)2Q 0
and smallenough, we havee Æ(t)w
(x)=e Æw
(x),
(t;x;a) =(t+;x;a); v
(t;x;a)=v(t+;x;a):
So, by bounding from below the left-hand sides of (56) and (60) by the
orresponding integrals performed on Q 0
, we get (38) and omplete the
proofof Proposition2..8. Inaddition,we haveestablished the onservation
of the kineti energy (17) and the averaged momentum equation (58).
3.4. Existene, uniqueness and partial regularity of
the pressure eld.
In this subsetion, we prove Theorem 2..9 in three steps : rst, we show
that the family (p
) is ompat in the sense of distributions in the interior
of Q, next, we establishadisrete version of (15), using divideddierenes
instead of derivatives, then we dedue the onvergene toward a unique
limit of (rp
) and nally we prove that the limitrp is a loally bounded
measure in the interior of Q.
Compatness in the sense of distributions.
From (53) with =0,we get, for all smooth vetor eld w parallelto D,
allsmooth funtion ompatly supported in ℄0;T[ and alljÆj1, that
Z
Q p
(t;y)(J(e Æ(t)w
(y)) 1)dydt 2
+C; (61)
where C depends on, w,and J denotes the Jaobiandeterminant. More-
over, we know that R
D p
(t;x)dx=0 holds true forall t2[0;T℄. Aording
to Moser's lemma (see [13℄ for a reent referene), by varying Æ, and w,
we an generate asuÆiently large set of smooth funtions ompatlysup-
ported in the interior of Q of form (t;x) !(J(e Æ(t)w
(y) 1) so that the
boundedness of the family (p
)in the sense of distributions isenfored.
More preisely : Let (t) and (x) be two ompatly supported smooth
funtions. We further assume
Z
(x)dx=0; supjj<1:
1)Set(s;x)=1+s(x); s2[0;1℄;
2)Solve
-=
withhomogeneousNeumannboundaryonditions;3)Set
v(s,x)=r(x)
(s;x);
sothat
s
+r(v)=0:
4)Solve
s
g(s;x)=v(s;g(s;x)); g(0;x)=x:
Then;wehave
det(D
x
g(s;x))=(s;x):
Thus
det(D
x
g((t);x))=1+(t)(x):
So;wehaveobtained;
sup
R
p
(t;x)(t)(x)dxdt<+1;
whihprovesthat
p
is bounded (up toanirrelevant addedfuntion of t only)in the sense of
distributions.
A disrete optimality equation.
Let (;m = v) be a xed arbitrarily hosen optimal solution. We all
test funtion on Q 0
(resp. on Q) any funtion f ontinuous on Q 0
(resp.
on Q), vanishing near t = 0 and t = T, with ontinuous derivatives in t
and x. Notie thatf(t;x;a) doesnotneessarilyvanishwhenx approahes
D. Let f be suh a test funtion with values in [0;1℄ and w be a smooth
divergene-free vetor eld on D, parallel to D. We will denote by C
f a
generionstantdependingonT,D,wandf onlythrough thesup normof
r
x
f,the L 2
(Q 0
;d)normof
t
f and the supportof f int. (It isimportant
that C
f
involves
t
f onlythrough its L 2
(Q 0
;d)norm.) Letus introdue
I = Z
1
Æ (
(t;e
Æw
(x);a)+
(t;x;a))f(t;x;a)d(t;x;a)
where
0is dened by
= (
t
+
1
2 jr
x
j
2
+p
): (62)
From (35) we dedue that
I 1
Æ Z
fd
2
Æ
: (63)
Wean split I intoI
1 +I
2 +I
3 with
I
1
= Z
1
Æ (p
(t;e
Æw
(x)) p
(t;x))fd;
I
2
= Z
1
2Æ [jr
x
j
2
(t;e Æw
(x);a) jr
x
j
2
(t;x;a)℄fd=
Z
1
2Æ [r
x
(t;e
Æw
(x);a) r
x
(t;x;a)℄:[r
x
(t;e
Æw
(x);a)+r
x
(t;x;a)℄fd
Z
1
Æ Z
[r
x
(t;e
Æw
(x);a) r
x
(t;x;a)℄:v(t;x;a)fd 1
Æ (Æ
2
+ 2
)C
f
(by Shwarz inequality and the resultatsof Proposition 2..8) and
I
3
= Z
1
Æ [
t
(t;e
Æw
(x);a)
t
(t;x;a)℄fd
= Z Z
1
0 d[
t r
x
(t;e
Æw
(x);a):w(e Æw
(x))℄fd=I
4 +I
5 +I
6
;
where, by using (14) and by integrating by part in t,
I
4
= Z Z
1
0 d
t fr
x
(t;e
Æw
(x);a):w(e Æw
(x))d
Z
t
fv:wd (Æ+)C
f
(fromtheresults ofProposition2..8 and Shwarz inequality,whihinvolves
the L 2
(Q 0
;d)norm of
t
f,taken into aountthrough C
f )
I
5
= Z
(v(t;x;a):r
x f)(r
x
(t;e
Æw
(x);a):w(e Æw
(x))dd:
Z
(v(t;x;a):r
x
f)(v(t;x;a):w(x))d (Æ+)C
f
(here C
f
involvesthe sup norm of r
x
f) and,
I
6
= Z
((v(t;x;a):r
x )[r
x
(t;e
Æw
(x);a):w(e Æw
(x))℄fdd:
Weget (after integrating in 2[0;1℄)
I
6
= 1
Æ Z
(v(t;x;a):r
x )[
(t;e
Æw
(x);a))
(t;x;a))℄fd=I
7 +I
8
;
where
I
7
= 1
Æ Z
v(t;x;a):[r
x
(t;e
Æw
(x);a)) r
x
(t;x;a))℄fd;
I
8
= 1
Æ Z
v(t;x;a):((r
x
(t;e
Æw
(x);a):r
x )(e
Æw
(x) x))fd
Z
v(t;x;a):((v(t;x;a):r
x
)w(x))fd ÆC
f :
Thus, sine I = I
1 +I
2 +I
3
(where I
3
= I
4 +I
5 +I
6
and I
6
= I
7 +I
8 )
satises (63), we get
Z
1
Æ (p
(t;e
Æw
(x)) p
(t;x))fd Z
v:(
t
+v:r
x
)(wf)d (64)
1
Æ (Æ
2
+ 2
)C
f :
Let t ! (t) a uto funtion [0;T℄, with values in [0;1℄, vanishing near
t = 0, t =T, and with value 1 on the support in t of f. Sine f takes its
values in [0;1℄,wean apply (64) to (1 f)= f insteadof f. We get
Z
1
Æ (p
(t;e
Æw
(x)) p
(t;x))( f)d+ Z
v:(
t
+v:r
x
)(w wf)d
1
Æ (Æ
2
+ 2
)C
f :
Both terms involving vanish, the rst one beause R
(t;x;da) = 1 and
e Æw
isLebesguemeasure-preserving, theseond onebeauseof(58). So, we
an put an absolutevalue on the left-hand side of (64), and, then, perform
the symmetry w! w to obtaina rst disreteversion of (15), namely
j Z
1
2Æ (p
(t;e
Æw
(x)) p
(t;e
Æw
(x)))fd Z
v:(
t
+v:r
x
)(fw)dj (65)
1
Æ (Æ
2
+ 2
)C
f :
Existene and uniqueness of the pressure gradient.
Ifweuse in(65) atest funtionf(t;x;a)=f(t;x)thatdoesnot depend on
a, we get, sine R
(t;x;da)=1,
j Z
1
2Æ (f(t;e
Æw
(y)) f(t;e Æw
(y)))p
(t;y)dtdy (66)
Z
v:(
t
+v:r
x
)(wf)dj 1
Æ (Æ
2
+ 2
)C
f
;
whereC
f
,now, dependsonf onlythrough the supportintof f andonthe
norm
jjjfjjj=sup
Q
(jfj+jr
x
fj)+( Z
Q j
t fj
2
dtdx) 1=2
: (67)
Sine, in the sense of distributions, the family (p
) is bounded, there are
luster points. Let p be any one of the luster points. Letting rst !0,
and then Æ ! 0, in (66), shows that p satises (40). Sine (;m = v)
an be any of the optimalsolutions, this shows the uniqueness of rp and,
therefore, the onvergene of the entire family (rp
) to rp in the sense of
distributions. Notie alsothat (66) impliesthe onvergene of
p
(t;e
Æw
(x)) p
(t;e
Æw
(x)) !p(t;e Æw
(x)) p(t;e Æw
(x)); (68)
as ! 0, Æ being xed, for the weak-* topology of the dual spae of the
separable Banah spae obtained by ompletion of the smooth funtions
ompatlysupported in℄0;T[D with respet tothe normdened by (67).
The pressure gradient is a measure.
To showTheorem 2..9, wehave nowto prove that rp isa loallybounded
measureintheinteriorofQ. Lett !(t)asmoothutofuntionon[0;T℄,
withvaluesin[0;1℄,0neart=0andt =T and1awayfromaneighborhood
of 0 and T. Let >0, Æ > 0 small enough so that t 2 (t ) vanishes
near 0 and T,for every in[ 1;+1℄. It is diÆulttoestimate
Z
Q (t)jp
(t;e
Æw
(x)) p
(t;x)jdxdt;
so weonsider the followingtime regularization
I = Z
Q (t)j
Z
1
0 (p
(t+;e Æw
(x)) p
(t+;x))djdxdt
= Z
Q 0
(t)j Z
1
0 (p
(t+;e Æw
(x)) p
(t+;x))djd(t;x;a)
(using R
A
(t;x;da) = 1). Let us onsider
(t;x;a) dened by (62), whih
is nonnegative and of whih the integral is bounded by 2
(aording to
(35) in Proposition 2..7). Wehave I I
1 +I
2 +I
3
, where
I
1
= Z
j Z
1
0 (
(t+;e Æw
(x);a)
(t+;x;a))djd;
I
2
= Z
j Z
1
0 (
t
(t+;e Æw
(x);a)
t
(t+;x;a))djd;
I
3
= Z
j Z
1
0 (
1
2 jr
x
j
2
(t+;e Æw
(x);a) 1
2 jr
x
j
2
(t+;x;a))djd:
The thirdterm iseasytohandle, thanksto(38) and Shwarz inequality. It
an be bounded by (+Æ+)C, where C, as all the oming up onstants
always denoted by C, depends only on D, T, and w. The seond term
an be bounded, after using the mean value theorem and integrating in ,
by
I
2 Æ
Z
Z
1
0 dj
1
[r
x
(t+;e Æ
(x);a) r
x
(t;e
Æ
(x);a)℄:w(e Æ
(x))jd
(herewe see thatthe timeregularizationavoids theuse of
t r
x
onwhih
we haveno ontrol)
C Æ
(+Æ+):
Let us now onsider the most deliate term, I
1
. We start with a rough
estimate, using
0 and R
d
2
, toobtain
I
1
Z
[
(t+;e Æw
(x);a)+
(t+;x;a)℄dd
2
2
+ Z
[
(t+;e Æw
(x);a)+
(t+;x;a) 2
(t;x;a)℄dd
=2 2
I
4 I
5 I
6
with
I
4
= Z
[
t
(t+;e Æw
(x);a)+
t
(t+;x;a) 2
t
(t;x;a)℄dd;
I
5
= 1
2 Z
[jr
x
(t+;e Æw
(x);a)j 2
+jr
x
(t+;x;a)j 2
2jr
x
(t;x;a)j 2
℄dd;
I
6
= Z
[p
(t+;e Æw
(x))+p
(t+;x) 2p
(t;x)℄dd:
Notie that we go bakward only apparently, sine now there is no ab-
solute value any longer in the integrals. In partiular I
6
vanishes sine
R
(t;x;da) = 1, e sw
is Lebesgue measure-preserving and R
p
(t;x)dx = 0.
The same bound is obtained for I
5
as for I
3
earlier. The treatment of I
4
needs more are. The idea is to get rid of the time derivatives by using
(14). Sine vanishesnear 0and T, therewillbenoboundaryterms om-
ingout ofthe integration by parts,however
t
has tobehandled arefully
beause wewish toinvolve
onlythrough itspartial derivatives. We split
I
4
=I
7 +I
8 with
I
7
= Z
[(t)(
t
(t+;e Æw
(x);a)+
t
(t+;x;a))
2(t )
t
(t;x;a)℄dd;
I
8
= Z
2((t ) (t))
t
(t;x;a)dd:
To treat I
8
,we have togo bakward again
I
8
= Z
2((t ) (t))(
(t;x;a))dd
Z
((t ) (t))jr
x
(t;x;a)j 2
dd Z
2((t ) (t))p
(t;x)dd:
The lastterm vanishes (sine R
(t;x;da)=1and R
p
(t;x)dx=0) and the
tworst onesareeasilybounded by C. Letusnowonsider I
7
=I
9 +2I
10 ,
where
I
9
= Z
(t)[
t
(t+;e Æw
(x);a)
t
(t+;x;a)℄dd;
I
10
= Z
[(t)
t
(t+;x;a) (t )
t
(t;x;a)℄dd:
Wehave
I
9
=Æ Z
(t) Z
1
0 d
t r
x
(t+;e Æw
(x);a):w(e Æw
(x))dd
= Æ
Z
(t) Z
1
0 d[r
x
(t+;e Æw
(x);a) r
x
(t;e
Æw
(x);a)℄:w(e Æw
(x))d;
whihan bebounded as I
2 by C
Æ
(+Æ+). We alsohave
I
10
= Z Z
1
0 d
t
[(t (1 ))
t
(t+;x;a)℄dd:
Here, we use (23) and get
Z Z
1
0
d(t (1 ))r
x
t
(t+;x;a):v(t;x;a)dd:
Thus I
10
=I
11 +I
12
, where
I
11
= Z Z
1
0 d
t
[(t (1 ))r
x
(t+;x;a)℄:v(t;x;a)dd
= Z
[(t)r
x
(t+;x;a) (t )r
x
(t;x;a)℄:v(t;x;a)dd;
I
12
= Z Z
1
0 d
0
(t (1 ))r
x
(t+;x;a):v(t;x;a)dd:
So, I
10
is bounded by (+Æ+)C.
Finally,beause (t)=1 away from t=0 and t =T, we have shown that,
for >0 smallenough,
Z
Q j
Z
1
0 (p
(t+;e Æw
(x)) p
(t+;x))djdxdt (69)
(1+Æ=)(+Æ+)C;
whereC depends onlyonD,T, and w. Letting !0, withfrozen =Æ,
in (69),shows that the limitp satises
1
Æ Z
<p(t;x);f(t Æ;e Æw
(x)) f(t Æ;x)>d Csupjfj
for every smooth funtion f(t;x) 0 with ompat support in0 <t <T.
Finally,we obtain, whenÆ !0,
<rp(t;x):w(x);f(t;x)>Csupjfj;
where C depends on D, T, w and the support of f, whih shows that rp
is a loally bounded measure in the interior of Q and ompletes the proof
of Theorem 2..9.
3.5. Optimality equations.
In this subsetion, we dedue (15) from (65) by letting Æ ! 0, whih
makes some problem sine rp
onverges to the measure rp only in the
senseofdistributions. In(65),letusassumethatthesupportoff isompat
in the interior of Q 0
. Then, given a xed vetor ein the unit ballof R d
, we
an hoose win suha way that, for jÆj small enough, e Æw
(x)=x+Æe and
w(x)=e holdtrue forall(t;x)suh thatf(t;x;a)6=0. Then (65)beomes
j Z
1
2Æ (p
(t;x+Æe) p
(t;x Æe))fd Z
v:(
t
+v:r
x
)(fe)dj (70)
1
Æ (Æ
2
+ 2
)C
f :
where C
f
depends onD,T, the normof f dened by (67) and the support
of f.
Let a radial nonnegative mollier on R d
, supported in the ball entered
at the origineand of radius 1=2, and e a xed vetor in the same ball. For
eah y in this ball, we perform the hange of variable e ! e+y in (70),
multiplyby (y)and integratewith respet to y. Then,weobtain
j Z
1
2Æ (p
(t;x+Æ(e+y)) p
(t;x Æ(e+y)))fd(y)dy
+ Z
v:e(
t
f +v:r
x
f)djÆC
f :
Sine isradial, we an replae the seond y by y and rewrite
j Z
1
2Æ (p
(t;x+Æe) p
(t;x Æe)f
;Æ;
(t;x)dtdx (71)
+ Z
v:e(
t
f +v:r
x
f)djÆC
f
;
where f
;Æ;
is dened by
f
;Æ;
(t;x)= Z
f(t;x Æy;a)(t;x Æy;da)(y)dy: (72)
This funtion belongs to the Banah spae obtained by ompletion of the
test funtions with respet to the norm dened by (67). Indeed, it is om-
patly supported in time, smooth in x and suÆiently smooth in t, sine,
beause of equation (14),
t f
;Æ;
(t;x)=
Z
(
t
+v(t;x Æy;a):r
x
)f(t;x Æy;a)(t;x Æy;da)(y)dy
r
x :
Z
v(t;x Æy;a)f(t;x Æy;a)(t;x Æy;da)(y)dy;
whih is Lebesgue square integrable on Q. Thus, for eah xed Æ, we an
take the limitof (71) when !0, by using (68), and obtain
j<rp(t;x):e;
Z
+1=2
1=2 f
;Æ;
(t;x 2Æe)d >+ Z
v:e(
t
f+v:r
x
f)dj (73)
ÆC
f
;
where<:;:>denotes the duality braketpairingmeasures and ontinuous
funtion on Q and the mean value theorem has been used to rewrite the
divided diereneas an integralin . Let usintrodue
Æ;e;
(t;x;a)= Z
+1=2
1=2 d
Z
(t;x 2Æe Æy;a)(y)dy; (74)
whihisamolliationof , ontinuous int andsmooth inxwith valuesin
the spae of Borel probability measures ina. (The smoothness in x omes
from the molliation by and the ontinuity in t from (14).) Denitions
(72) and (74) show that
j Z
+1=2
1=2 f
;Æ;
(t;x 2Æe)d Z
f(t;x;a)
Æ;e;
(t;x;da)j C
f Æ;
so wean rewrite(73) as
j<rp(t;x):e;
Z
f(t;x;a)
Æ;e;
(t;x;da) >+ Z
v:e(
t
f +v:r
x
f)dj (75)
ÆC
f :
Sine rpisaloallybounded measure inthe interiorof Qand
Æ;e;
(t;x;a)
is aprobability measure in a, we get
j<rp(t;x):e;
Z
f(t;x;a)
Æ;e;
(t;x;da)>j (76)
<jrp(t;x):ej;sup
a2A
jf(t;x;a)j>
Thus, the
Æ;e;
an beseen asaboundedsubset ofL 1
(jrpj;C(A)),dened
asthe dual spaeof the (separable) Banah spae L 1
(jrpj;C(A)),made of
allfuntionsf(t;x;a)whiharejrpjintegrablein(t;x)withvaluesinC(A),
the spaeofontinuousfuntionsina. Sothis familyisweak-*sequentially
relativelyompat. Beauseof(75),foreahxedeand,thereisaunique
possible luster point, whenÆ !0,denoted by
e;
and satisfying
<rp(t;x):e;
Z
f(t;x;a)
e;
(t;x;da)>=
Z
v:e(
t
f +v:r
x
f)d: (77)
Sine the right-hand side does not depend on and depends linearly on e,
the limit la limite
e;
annot depend on and e. Thus, this limit an be
denoted by and satises
<rp(t;x):e;
Z
f(t;x;a)(t;x;da) >=
Z
v:e(
t
f +v:r
x
f)d: (78)
Thus, we dedue from denition (74) that (19) holds true for the weak-
* topology of the dual spae of L 1
(jrpj;C(A)) and we an use it as the
denition of as the extension of with respet to the measure dtdx+
jrp(t;x)j. This ompletes the obtention of (15) and the proof of Theorem
2..3.
3.6. Consisteny of the relaxed problem with loal
smooth solutions of the Euler equations.
In this subsetion, we prove Theorem 2..4. Given a loal smooth solution
(u;p)ofthe Euler equations,itisenoughtoshowthat anyadmissiblesolu-
tion(;m) satisfyingK(;m) 1=2 R
Q juj
2
neessarilyisthepairassoiated
with u through (21),(22).
Let usonsider
f(t;x;a)=
t
g(t;i(a)):(x g(t;i(a)))
whih is a smooth funtion of (t;x;a) 2 Q 0
and satises f(T;h(i(a));a) =
f(0;i(a);a)=0. Thus, we an use (23) and get
Z
v:r
x fd=
Z
t fd:
So,
Z
v(t;x;a):
t
g(t;i(a)))d(t;x;a)
= Z
( 2
t
g(t;i(a)):(x g(t;i(a))) j
t
g(t;i(a))j 2
)d(t;x;a):
Sine (u;p)is asmooth solution tothe Euler equations,we have
2
t
g(t;i(a))= (r
x
p)(t;g(t;i(a))):
Thus
Z
1
2
jv(t;x;a)
t
g(t;i(a))j 2
d(t;x;a)
= Z
[ 1
2
(jv(t;x;a)j 2
j
t
g(t;i(a))j 2
)
(r
x
p)(t;g(t;i(a))):(x g(t;i(a)))℄d(t;x;a):
But, by assumption,
K(;m) 1
2 Z
Q juj
2
dtdx
where the right-handside is
Z
1
2 j
t
g(t;i(a))j 2
dtda= Z
1
2 j
t
g(t;i(a))j 2
d(t;x;a)
(sine R
(t;dx;a)= R
(0;dx;a)=1,whihfollows fromd'apres (23)). So,
Z
1
2
jv(t;x;a)
t
g(t;i(a))j 2
d(t;x;a)
Z
( r
x
p)(t;g(t;i(a))):(x g(t;i(a)))d(t;x;a):
By denition of asthe largest eigenvalue of the seond spae derivatives
of p, and by the mean value theorem, we have
p(t;x) p(t;g(t;i(a))) (r
x
p)(t;g(t;i(a))):(x g(t;i(a)))
1
2
jx g(t;i(a))j 2
:
Sine
Z
(p(t;x) p(t;g(t;i(a))))d(t;x;a)=0
(indeed g is volume preserving and R
(t;dx;a) = 1, R
(t;x;da) = 1), we
have nallyobtained
Z
(jv(t;x;a)
t
g(t;i(a))j 2
jx g(t;i(a))j 2
)d(t;x;a)0: (79)
SineT 2
<
2
,the proofoftheorem2..4 isompleted,onewenotiethat
Z
jx g(t;i(a))j 2
d(t;x;a)=0
follows from the following generalization of the lassial one dimensional
Poinare inequality:
Proposition 3..2 Let t2[0;T℄!z(t)2R d
an absolutely ontinuous path
suththat R
T
0 jz
0
(t)j 2
dt<+1. Let(;m)beapairof respetivelynonnegative
and vetor-valued measures on Q = [0;T℄ R d
, suh that m = v, with
v 2L 2
(Q;d) d
, and
Z
(
t
+v:r
x
)d=(T;z(T)) (0;z(0)) (80)
for allsmooth funtions on Q. Then
Z
(jv(t;x) z 0
(t)j jx z(t)j 2
)d(t;x)0; (81)
when T 2
<
2
.
Proof of Proposition 3..2.
Set z(t)=z(0)for t0 and x >0. Introdue
(t;x)=inf Z
t
1
2 [j
0
()j 2
j()j 2
℄d; (82)
where the inmum is performed on all paths t 2 [ ;T℄ ! (t)2 R d
suh
that
( )=0; (t)=x z(t):
By the lassialone-dimensional Poinareinequality,
Z
t
(j
0
()j 2
j()j 2
)d 0;
for all paths suh that (t) = 0 for t = and t = T, provided that
( +T) 2
2
, whih holds true if > 0 is hosen small enough. We
dedue
(t;z(t))=0
(beause the inmum is ahieved by = z). By standard alulus, is a
C 1
solutionon Qof the Hamilton-Jaobiequation [3℄
t +
1
2 jr
x +z
0
j 2
+ 1
2 [jz
0
j 2
+jx zj 2
℄=0: (83)
(We write, for smallenough>0,
(t;x)=inf
y;
[(t ;x y)+ Z
t
t 1
2 [j
0
()j 2
j()j 2
℄d;
where the inmum is taken over all y 2 R d
and all paths starting from
x z(t) y attime t andreahingx z(t) attime t. Then, weperform
a Taylor expansion about = 0.) By applying (80) to , whih satises
(T;z(T))=(0;z(0)) =0, and by using (83), we get
Z
(v:r 1
2
[jr+z 0
j 2
jz 0
j 2
+jx zj 2
℄)d=0:
By this quantity exatlyis
1
2 Z
(jv z 0
j 2
jx zj 2
jv r z
0
j 2
)d;
whihompletes the proof of (81), Proposition 3..2 and Theorem 2..4.
3.7. Limit behaviour of solutions as goes to 0.
In this subsetion we prove Proposition 2..5. Let us onsider an solution
(asdenedinDenition1..1),andtheassoiatedmeasures(
;m
)(dened
