Stability of Two-dimensional Viscous Incompressible Flows Under Three-dimensional Perturbations and Inviscid Symmetry Breaking

0.15 MB·English

by  Claude Bardos

STABILITY OF TWO-DIMENSIONAL VISCOUS INCOMPRESSIBLE
FLOWS UNDER THREE-DIMENSIONAL PERTURBATIONS AND
INVISCID SYMMETRY BREAKING
2
1
0 CLAUDEBARDOS,MILTONC.LOPESFILHO,DONGJUANNIU,
2 HELENAJ.NUSSENZVEIGLOPES,ANDEDRISSS.TITI

n
a
ABSTRACT. Inthisarticleweconsiderweaksolutionsofthethree-dimensional
J
incompressiblefluidflowequationswithinitialdataadmittingaone-dimensional
9
symmetry group. We examine both the viscous and inviscid cases. For the
1
case of viscous flows, we prove that Leray-Hopf weak solutions of the three-

dimensional Navier-Stokesequations preserve initiallyimposedsymmetry and
]
P thatsuchsymmetricflowsarestableundergeneralthree-dimensionalperturba-
A tions, globally in time. We work in three different contexts: two-and-a-half-
. dimensional, helicalandaxi-symmetricflows. Intheinviscidcase,weobserve
h that, as a consequence of recent work by De Lellisand Sze´kelyhidi, there are
t
a genuinely three-dimensional weak solutions of the Euler equations with two-
m dimensionalinitialdata.Wealsopresenttwopartialresultswhererestrictionson
thesetofinitialdata,andonthesetofadmissiblesolutionsruleoutspontaneous
[
symmetrybreaking;oneisduetoP.-L.Lionsandtheotherisaconsequenceof

2 ourviscousstabilityresult.
v
2
4
7 MSCSubjectClassifications: 35Q35,65M70.
2 Keywords: Navier-Stokes equations, Euler equations, Leray-Hopf weak solu-
.
1 tions,helicalsymmetry,uniqueness ofweaksolutions, axi-symmetric flow.
0
2
1 1. INTRODUCTION
:
v Inthisarticle,weconsider theequations forincompressible fluidmotion:
i
X
ar ∂tu+(u·∇)u= −∇p+ν∆u+f, (1.1)
divu = 0,
(cid:26)
supplemented by appropriate initial and boundary data. Above, u = (u ,u ,u )
1 2 3
is the fluid velocity and p is the scalar pressure. The external force f and the
kinematic viscosity ν ≥ 0 are given. System (1.1) is referred to as the Navier-
Stokes equations in the viscous case (ν > 0), and as the Euler equations of ideal
fluidmotionintheinviscid case(ν = 0).
Let u = u(t,x) = u(t,x ,x ,x ) be a Leray-Hopf weak solution (see Defini-
1 2 3
tion2.1)oftheNavier-Stokes equations (1.1)forsomeν > 0inadomainΩ,with
zeroforcing. AssumethatthedomainΩ ⊂ R3 andtheinitialvelocityu ≡ u(0,·)
0
are symmetric with respect to a one-parameter group which is invariant under the
Date:January18,2012.
1
2 C.BARDOS,M.C.LOPESFILHO,D.NIU,H.J.NUSSENZVEIGLOPES,ANDE.S.TITI
Navier-Stokes evolution. For example, one may think of flow in the full three-
dimensional space, which is periodic in all three directions, for which the initial
velocity is periodic and invariant under vertical translations, i.e., whose compo-
nentsdonotdependontheverticalvariable. Ourmainnewresultisglobal-in-time
stability in the energy space of solutions which preserve the symmetry, within the
class of Leray-Hopf weak solutions of the three-dimensional Navier-Stokes equa-
tions. As a consequence, any Leray-Hopf weak solution of the three dimensional
Navier-Stokes equations which starts symmetric will stay symmetric for positive
time, ruling out spontaneous symmetry breaking within this class of weak solu-
tions. Wewillalso seethat, asaspecial case ofaconstruction duetoC.DeLellis
andL.Sze´kelyhidi,spontaneoussymmetrybreakingdoesoccuramongweaksolu-
tionsofthethree-dimensional Eulerequations.
Our analysis, in the viscous case, is closely related to weak-strong uniqueness
resultsfortheLeray-Hopfweaksolutionsoftheincompressible 3DNavier-Stokes
system,asubjectwithanold,largeanddeepliterature. Symmetricflows,regarded
asaspecialclassofthree-dimensionalflows,aremoreregularthanageneralthree-
dimensional Leray-Hopf weak solution. The idea behind weak-strong uniqueness
istoimposeadditionalregularityassumptionsonagivenweaksolutioninorderto
guarantee itisunique. Ourpoint of departure iswhether this additional regularity
ofsymmetricweaksolutions isenoughtoensureuniqueness.
Thefirstweak-strong uniqueness resultforLeray-Hopfsolutions oftheNavier-
Stokes equations is due to Sather and Serrin, see [18], and itis usually referred to
asSather-Serrin Uniqueness Criterion, see also the work ofG.Prodi [16]. Briefly
stated, a weak solution in Lq((0,T),Lp(Ω)) is unique if 3/p + 2/q = 1, 3 <
p < ∞. Recently this criteria was extended to the limit case p = 3, q = ∞, see
[17,3,4].
Generaltwo-dimensionalflows,forexample,areinL∞((0,T);L2)∩L2((0,T);H1),
which, by interpolation and Sobolev imbedding, are in Lq((0,T);Lp(Ω)) with
2 ≤ p < ∞, 2 ≤ q < 2p/(p − 2) or (p,q) = (2,∞). We call this region of
theextended (p,q)-plane R. Thehyperbola 3/p+2/q = 1lies strictly above the
regionR,approaching Ronlyas(p,q) → (∞,2). Hence,theSather-Serrin crite-
rion (or its extension tothe limit case p = 3, q = ∞)does not ensure uniqueness
oftwo-dimensionalflows,whenviewedasthree-dimensional flows. Theterminol-
ogytwo-dimensionalflows,inthiswork,meansthatthecomponentsofthevelocity
fields do not depend on the vertical variable, x . We observe that, depending on
3
the context, the velocity fields of two-dimensional flows can have either two or
threecomponents. Wealsorecallthattwo-dimensional flowsaresometimescalled
two-and-a-half-dimensional flows(denoted21Dflows)whenthevelocityfieldhas
2
threenon-trivial components (see,e.g.,Section2.3.1of[13]).
There is a large literature dedicated to extensions of the Sather-Serrin crite-
rion, see [5] and references therein. However, the results which have been ob-
tained tend to obey the same scaling as the Sather-Serrin condition. The prob-
lems treated in the current paper are, in a sense, off-scale, and, therefore, only
the extensions which have been obtained near the critical case (∞,2) are poten-
tially relevant to our work. One particularly noteworthy result was established by
STABILITYOF2DFLOWSUNDERTHREE-DIMENSIONALPERTURBATIONS 3
H. Kozono and Y. Taniuchi, see [8] and it concerns extending the Sather-Serrin
uniqueness criterion to vector fields which are bounded in L∞((0,T);L2(R3))∩
L2((0,T);BMO(R3)). In fact, vector fields which are in L2((0,T;H˙1(R2)),
suchasthesolutions ofthetwo-dimensional Navier-Stokes equations, areactually
boundedinL2((0,T);BMO(R3))becauseH˙1(R2) ⊂ BMO(R2)⊂ BMO(R3)
(we note emphatically that these vector fields are independent of the third vari-
able). However, these vector fields are not square-integrable in R3, so we cannot
use Kozono and Taniuchi’s criterion to address uniqueness (or stability) of two-
dimensional solutions viewedasthree-dimensional flows.
The original argument in [18] was formulated in an arbitrary domain, but, as
in Kozono and Taniuchi’s result, the extensions have been full-space results mak-
ing use of harmonic analysis machinery. Of course, to circumvent the fact that
two-dimensional flowsare not square integrable infull space, oneshould look for
uniqueness among 3D flows in another domain, such as flows which are periodic
inthethird variable. Itseemslikely that onecould adapt theproof ofKozono and
Taniuchi’s criterion to flows which are periodic in the third variable, and then ob-
tainanuniquenessandstabilityresultalongthelinessuggestedabove. However,in
thiswork,wewouldliketotakeamoreelementary approach, closertoSatherand
Serrin’soriginalargument,whichworksonaverticallyperiodicflowincylindrical
domainofgeneralshape,andinothersituations aswell.
Theproblemofstabilityoftwo-dimensionalflowsunderthree-dimensional per-
turbationsisverynaturalandinterestingfromthephysicalpointofview,andithas
beenthesubject ofprevious work. Thefirstresults inthis direction wereobtained
by G. Ponce, R. Racke, T.C. Sideris and E.S. Titi, see [15]. Their main result is
globalexistenceofastrongsolutionwhichstartsclose,inH1toatwo-dimensional
solution, alsoastabilityestimate. Theirresultwaslaterimprovedin[7,14],byre-
laxingregularityconditions ontheperturbation, butalwaysworkingintheclassof
strong solutions, and therefore, focusing their concern on global existence, rather
thanstability. Ourworkmayberegarded asanextension ofthese articles toweak
Leray-Hopfsolutions.
We are going to prove uniqueness and stability results for Leray-Hopf weak
solutions inthreedifferentcontexts:
(i) two-dimensional flow in an infinite straight cylinder with bounded and
smoothcross-section,withno-slipboundaryconditionandthree-dimensional
perturbations whichareperiodic intheverticaldirection;
(ii) helicalflowinastraightcircularcylinder, withno-slipboundarycondition
and general three-dimensional perturbations with the same period as the
helicalflow;
(iii) axi-symmetric flow in the interior of an axi-symmetric torus with smooth
cross-section bounded away from the symmetry axis, no-slip boundary
condition andageneral threedimensional perturbation.
Ineachcase,existenceofasymmetricweaksolutionfortheNavier-Stokesequa-
tions whentheinitial data issymmetric isanimplicit requirement ofour analysis,
and can be obtained by an easy adaptation of the classical argument by Leray.
4 C.BARDOS,M.C.LOPESFILHO,D.NIU,H.J.NUSSENZVEIGLOPES,ANDE.S.TITI
Globalwell-posedness ofweaksolutions isalsoknowninallthree cases. Weem-
phasize that these well-posedness results allrefer tothecorresponding symmetry-
reduced equations. Forcase (i), it was pointed out in Proposition 2.7 of [13], that
global existence of two-dimensional flows, regarded as three-dimensional flows,
reduces to the global well-posedness result of weak solutions of the standard 2D
Navier-Stokes equations in a bounded domain, which was established in [9]. For
globalexistenceanduniqueness ofweaksolutionsincase(ii)see[12]and,incase
(iii),see[10,20].
Concerning the inviscid case, we discuss three results. The first result is exis-
tenceofagenuinely3Dweaksolutionofthe3DEulerequationsinaperiodiccube,
withtwo-dimensional initial data. Theresult isaspecial case ofaconstruction by
De Lellis and Sze´kelyhidi in [1], as formulated by Wiedemann in [21]. The sec-
ond result is a consequence of a weak-strong uniqueness theorem for dissipative
solutions of the Euler equations, due to P.-L. Lions in [11]. The third result is a
corollary of our viscous stability estimates, applied to Euler solutions which are
vanishing viscosity limitsinaspecificway.
The remainder of this work is divided into four sections. Section 2 contains
basic definitions and notation, Section 3 concerns the viscous results, Section 4
containstheinviscid results, andSection5isfinalcommentsandconclusions.
2. PRELIMINARIES
Inthis section wefixnotation andsetdownsome basic definitions. Inthis arti-
cle,weareconcerned withincompressible flowsinthree contexts -triplyperiodic
flow in a box, flow in an infinite vertical cylinder whose horizontal cross sections
arebounded andsmoothandwhichareperiodic inthevertical direction andflows
inabounded axi-symmetric domain. Todiscuss theincompressible Navier-Stokes
equationsinthesecontextswefirstintroducetheHilbertspacesusuallydenotedby
H andV intheliterature,adjustingthingsaccordingtothespecificcaseasfollows:
(1) for the periodic box Ω ≡ (0,1)3, the spaces H(Ω) and V(Ω) are the clo-
sureoftheC∞,periodic,divergence-free vectorfieldsinΩwithrespectto
theL2 andH1 normsinΩ,respectively.
(2) fortheperiodiccylinderΩ = D×(0,L),whereL > 0andDisabounded
smooth domain in R2, the spaces H(Ω) and V(Ω) are the closure of the
C∞ vector fields which are, periodic in the vertical variable, compactly
supported inthehorizontal sections and divergence-free inΩwithrespect
totheL2 andH1 normsinΩ,respectively.
(3) for a smooth axi-symmetric domain Ω, the spaces H(Ω) and V(Ω) are
the closure of the C∞ vector fields which are, compactly supported and
divergence-free in Ω with respect to the L2 and H1 norms in Ω, respec-
tively.
We denote by D(Ω)the space of C∞ test functions, periodic in the case of the
cube, compactly supported for the axi-symmetric domain and and periodic in the
vertical variable, compactly supported in the horizontal direction for the periodic
STABILITYOF2DFLOWSUNDERTHREE-DIMENSIONALPERTURBATIONS 5
cylinder. WewillalsousethenotationH (Ω)forthevectorspaceH(Ω)endowed
w
withtheweaktopology inL2.
Letusrecall the definition ofaLeray-Hopf weak solution ofthe Navier-Stokes
system:
Definition 2.1. Let Ω be either the periodic box, the periodic cylinder or an axi-
symmetricdomainasaboveandletu∈ C0([0,∞);H (Ω))∩L∞((0,∞);H(Ω))∩
w
L2 ([0,∞);V(Ω)). ThenuisaweakLeray-Hopfsolutionof(1.1)withinitialdata
loc
u ∈ H(Ω)andforcing f ∈ L2((0,T);H−1(Ω))if:
0
(1) foranytestfunctionφ ∈C∞([0,∞);D(Ω))suchthat divφ= 0wehave:
c
∞
{−∂ φ·u−[(u·∇)φ]·u+ν∇u: ∇φ} dxdt− u ·φ(0,x)dx
t 0
Z0 ZΩ ZΩ
∞
= hφ(t,·),f(t,·)idt,
Z0
where A : B ≡ a b is the trace product of two matrices and h·,·i
i,j ij ij
denotes theduality pairingbetweenH1(Ω)andH−1(Ω).
P 0
Additionally,
(2) foranyt > 0,
t t
ku(t,·)k2 +2ν k∇u(s,·)k2 ds ≤ ku k2 +2 hu(s,·),f(s,·)ids.
L2(Ω) L2(Ω) 0 L2(Ω)
Z0 Z0
(2.1)
Note that by density arguments, and the continuity of the terms used in the
identity in item (1) of Definition 2.1, one can extend the Definition 2.1 to allow
for the use of test functions φ ∈ C∞([0,∞);V(Ω)) in the identity in item (1) of
c
Definition2.1.
We also require a definition of weak solution for the Euler equations, but only
inthecaseoftheperiodic boxandwithoutforcing, see[1,21].
Definition2.2. LetΩ = (0,1)3betheperiodicboxandletu ∈ C0([0,∞);H (Ω)).
w
Wesay that uis a weak solution of the Euler equations ((1.1), ν = 0) with initial
velocity u ∈ H(Ω) if for any test function φ ∈ C∞([0,∞);D(Ω)) such that
0 c
divφ = 0wehave:
∞
{∂ φ·u+[(u·∇)φ]·u} dxdt+ u ·φ(0,x)dx = 0.
t 0
Z0 ZΩ ZΩ
3. VISCOUS FLOW
Inthissection wewillstateandprovestability results forLeray-Hopf weakso-
lutionsoftheNavier-Stokesequations(1.1),withzeroforcing,inthethreecontexts
described intheintroduction.
We start with three-dimensional perturbations of two-dimensional flows. Re-
callthattwo-dimensional flowsrefertosolutions ofthethree-dimensional Navier-
Stokesequations whichareindependent ofx .
3
6 C.BARDOS,M.C.LOPESFILHO,D.NIU,H.J.NUSSENZVEIGLOPES,ANDE.S.TITI
Theorem3.1. LetD ⊂ R2beaboundeddomainwithsmoothboundary. Consider
u ∈ H(D) and let u ∈ C0([0,∞);H (D))∩L2((0,∞);V(D)) be the unique
0 w
weak solution of the 21D incompressible Navier-Stokes equations having, as ini-
2
tial data, u . Fix L > 0 and set C = D×(0,L). Let v ∈ L∞((0,∞);H(C))∩
0
L2((0,∞);V(C))beaLeray-Hopfweaksolutionofthethree-dimensional incom-
pressible Navier-Stokes equations with initial data v , where v ∈ H(C). The
0 0
followingestimateholdstrue:
27
kv−uk2 (t) ≤ kv −u k2 exp ku k4 , forallt ≥ 0.
L2(C) 0 0 L2(C) 64ν4 0 L2(D)
(cid:18) (cid:19)
ProofofTheorem3.1. We begin by following the argument in the proof of Theo-
rem6in[18]. FixT > 0andletη beastandard1-dimensional mollifier(smooth,
ε
non-negative, even,supported in(−ε,ε)andwithunitintegral). Define
T
uε = uε(t,x) ≡ η (t−s)u(s,x)ds.
ε
Z0
Defineinananalogous mannervε = vε(t,x).
Then, using uε as test function in the weak formulation of the equation for v,
andvε astestfunction fortheequation foru,wefindthefollowingtwoidentities:
T T
−(u,vε)(T)+ (u,∂ vε)ds−ν (∇u,∇vε)ds (3.1)
t
Z0 Z0
T
= − ((u·∇)vε,u)ds−(u ,vε);
0 0
Z0
T T
−(v,uε)(T)+ (v,∂ uε)ds−ν (∇v,∇uε)ds (3.2)
t
Z0 Z0
T
= − ((v·∇)uε,v)ds−(v ,uε),
0 0
Z0
where(·,·)denotestheinnerproductinL2(C).
Weaddthesetwoidentities,usingthefactthat T(u,∂ vε)ds = − T(v,∂ uε)ds,
0 t 0 t
andwefind
R R
T
−(u,vε)(T)−(v,uε)(T)−ν [(∇u,∇vε)+(∇v,∇uε)]ds (3.3)
Z0
T
= − [((u·∇)vε,u)+((v·∇)uε,v)]ds−(u ,vε)−(v ,uε).
0 0 0 0
Z0
Wemultiply(3.3)by2andletε → 0toobtain
T
−2(u,v)(T)−4ν (∇u,∇v)ds (3.4)
Z0
T
= 2 (((v −u)·∇)(v−u),u)ds−2(u ,v ).
0 0
Z0
STABILITYOF2DFLOWSUNDERTHREE-DIMENSIONALPERTURBATIONS 7
Next, we use the energy inequality, satisfied by both u and v (see Definition
2.1):
T
kuk2 (T)+2ν k∇u(s,·)k2 ds ≤ ku k2 ; (3.5)
L2(C) L2(C) 0 L2(C)
Z0
T
kvk2 (T)+2ν k∇v(s,·)k2 ds ≤ kv k2 . (3.6)
L2(C) L2(C) 0 L2(C)
Z0
Introduce w ≡ v−uandadd(3.4),(3.5)and(3.6)tofind
T T
kwk2 (T)+2ν k∇wk2 ds ≤ kw k2 +2 (w·∇w,u)ds. (3.7)
L2(C) L2(C) 0 L2(C)
Z0 Z0
This isprecisely inequality (27) in [18]. Atthis point wedepart from the argu-
mentpresented in[18]anduse thefact thatuistwo-dimensional. Weanalyze the
nonlinear termusingthetwo-dimensional Ladyzhenskaya inequality inD:
T T L
((w·∇)w,u)ds = [(w·∇)w]·udx dx dx ds
1 2 3
Z0 Z0 Z0 ZD
T L
≤ kwkL4(D)k∇wkL2(D)kukL4(D)dx3ds
Z0 Z0
T L
≤ 21/4 kwkL1/22(D)k(∂x1,∂x2)wk1L/22(D)k∇wkL2(D)kukL4(D)dx3ds
Z0 Z0
T L
≤ 21/4 kwk1L/22(D)k∇wk3L/22(D)kukL4(D)dx3ds
Z0 Z0
T L 27 T L
≤ ν k∇wk2 dx ds+ kwk2 kuk4 dx ds,
L2(D) 3 128ν3 L2(D) L4(D) 3
Z0 Z0 Z0 Z0
byYoung’sinequality. Therefore, usingthefactthatku(s,·)k4 isindependent
L4(D)
ofx ,weobtain
3
T T 27 T
((w·∇)w,u)ds ≤ ν k∇wk2 ds+ kuk4 kwk2 ds.
L2(C) 128ν3 L4(D) L2(C)
Z0 Z0 Z0
(3.8)
Weinput(3.8)in(3.7)tofind
27 T
kwk2 (T)≤ kw k2 + kuk4 kwk2 ds. (3.9)
L2(C) 0 L2(C) 64ν3 L4(D) L2(C)
Z0
Therefore, byGronwall’sLemmawededucethat
27 T
kwk2 (T) ≤ kw k2 exp kuk4 ds . (3.10)
L2(C) 0 L2(C) 64ν3 L4(D)
(cid:18) Z0 (cid:19)
Finally,weuseagaintheLadyzhenskaya inequality toestimate:
8 C.BARDOS,M.C.LOPESFILHO,D.NIU,H.J.NUSSENZVEIGLOPES,ANDE.S.TITI
T T
kuk4 ds ≤ 2 kuk2 k∇uk2 ds
L4(D) L2(D) L2(D)
Z0 Z0
≤ 2kuk2 k∇uk2 ,
L∞((0,T);L2(D)) L2((0,T);L2(D))
which, using (3.5)together withthefact thatuisindependent ofx yields thede-
3
siredresult,oncewereplaceT byanarbitraryt ≥ 0andnoticethatthedependence
onLcancelsout. (cid:3)
Remark 3.1. An immediate corollary of Theorem 3.1 is the uniqueness of Leray-
Hopfweaksolutions fortwo-dimensional initialdata.
Next,wewillexamineavariantofTheorem3.1,pertaining tohelicalflows.
A vector field U is called helical, with step σ ∈ R\{0} if, for any θ ∈ R and
anyx ∈ R3,
cosθ sinθ 0 0
U −sinθ cosθ 0 x+ 0 =U(x).
   
0 0 1 σ θ
2π
   
Werefer the reader to [12] for well-posedness results for the Navier-Stokes equa-
tions with helical symmetry. For simplicity, we will focus on the special case of
helicalflowsinastraightcircularpipe.
Theorem 3.2. LetD bethe unit disk intheplane, while C denotes theunit cylin-
der D × (0,1). Let u ∈ H(C) be a helical vector field with step equal to 1.
0
Let u ∈ C0([0,∞);H (C)) ∩ L2((0,∞);V(C)) be the unique weak solution
w
of the helical incompressible Navier-Stokes equations having, as initial data, u ,
0
given in Theorem 3.3 of [12]. Let v ∈ H(C) and let v ∈ C0([0,∞);H (C))∩
0 w
L2((0,∞);V(C))beaLeray-Hopfweaksolutionofthethree-dimensional incom-
pressibleNavier-Stokesequationswithinitialdatav . Then,thefollowinginequal-
0
ityisvalid:
27
kv−uk2 (t)≤ kv −u k2 exp ku k4 , forallt ≥ 0.
L2(C) 0 0 L2(C) 64ν4 0 L2(C)
(cid:18) (cid:19)
ProofofTheorem3.2. We can use the same proof as for the 21D case once we
2
makethefollowingobservations:
(i)theLp(D)-normsofuareindependent ofx ,foranyp ≥ 1;
3
(ii)theL2(D)-normof(∂ ,∂ )uisindependentofx andbounded
x1 x2 3
abovebytheL2(C)-normof∇u.
(cid:3)
Remark3.2. Asbefore,thiseasilyyieldsuniquenessofLeray-Hopfweaksolutions
withhelicalinitialdata.
Lastly,wediscussthecaseofaxi-symmetricflows.
STABILITYOF2DFLOWSUNDERTHREE-DIMENSIONALPERTURBATIONS 9
Theorem3.3. LetDbeabounded,smoothdomaincompactlycontainedin{(r,z) |0 <
r < ∞, z ∈ R} and set C = {(r,z,θ) | (r,z) ∈ D, 0 ≤ θ ≤ 2π}. Let
u ∈ H(C) be an axially symmetric vector field. Let u ∈ C0([0,∞);H (C))∩
0 w
L2((0,∞);V(C)) be the unique weak solution of the axi-symmetric incompress-
ible Navier-Stokes equations having, as initial data, u , given in [10, 20]. Let
0
v ∈ H(C)and let v ∈ C0([0,∞);H(C))∩L2((0,∞);V(C)) bea Leray-Hopf
0
weak solution of the three-dimensional incompressible Navier-Stokes equations
with initial data v . There exists a constant M = M(D,ν) > 0 such that the
0
followinginequality isvalid:
kv−uk2 (t)≤ kv −u k2 exp Mku k4 , forallt ≥ 0.
L2(C) 0 0 L2(C) 0 L2(C)
(cid:16) (cid:17)
ProofofTheorem3.3. Wemustmakesmallmodificationsoftheproofforthe21D
2
2π
case, beginning by writing the integral over C as with respect tothe mea-
0 D
surerdrdzdθ.
R R
Weestimatethenonlinear termasfollows:
T T 2π
((w·∇)w,u)ds = [(w·∇)w]·urdrdzdθds
Z0 Z0 Z0 ZD
T 2π
≤ kwkL4(D,rdrdz)k∇wkL2(D,rdrdz)kukL4(D,rdrdz)dθds
Z0 Z0
T 2π
1/2 1/2
≤ K kwkL2(D,rdrdz)k(∂r,∂z)wkL2(D,rdrdz)k∇wkL2(D,rdrdz)kukL4(D,rdrdz)dθds,
Z0 Z0
where K > 0 is a constant appearing in the two-dimensional Ladyzhenskaya in-
equality in D, valid since w vanishes on the boundary of D for each fixed θ, to-
gether with the fact that D is bounded away from the axis of symmetry, so that
r > a,forsomefixeda > 0;
T 2π
1/2 3/2
≤ K kwkL2(D,rdrdz)k∇wkL2(D,rdrdz)kukL4(D,rdrdz)dθds
Z0 Z0
T 2π T 2π
≤ ν k∇wk2 dθds+K kwk2 kuk4 dθds,
L2(D,rdrdz) L2(D,rdrdz) L4(D,rdrdz)
Z0 Z0 Z0 Z0
forsomeK > 0,resulting fromusingYeoung’sinequality,
T T 1
e≤ k∇wk2L2(C)ds+K 2πkuk4L4(C)kwk2L2(C)ds,
Z0 Z0
where we have used the fact that ku(e·,θ)k4L4(D,rdrdz) is independent of θ and is
equalto(1/2π)kuk4 . Weobservethat,above,theconstant K dependsona.
L4(C)
BytheGronwallLemmawededuce, asbefore,that
K T
kwk2 (T) ≤kw k2 exp kuk4 ds .
L2(C) 0 L2(C) 2π L4(C)
Z0 !
e
Finally, we use again the two-dimensional Ladyzhenskaya inequality for u,
noticing that the derivatives which appear are with respect to r and z and, hence,
10 C.BARDOS,M.C.LOPESFILHO,D.NIU,H.J.NUSSENZVEIGLOPES,ANDE.S.TITI
their L2(D,rdrdz)-norms are independent of θ. This, together with the energy
inequality (3.5), yields the desired result, replacing T by an arbitrary time t ≥ 0.
Thisconcludes theproof.
(cid:3)
Globalexistenceanduniquenessofweaksolutionsfortheaxi-symmetricNavier-
StokesequationswasestablishedbyO.Ladyzhenskaya, see[9],butonlyunderthe
assumption that the axi-symmetric fluid domain be bounded away from the sym-
metry axis, i.e., r > a, for some a > 0. This restriction has the same origin as
inTheorem 3.3,namely, loss ofessential 2Dscaling atthesymmetryaxis. (Addi-
tionalresultsonglobalregularityofspecialsolutionsoftheaxi-symmetricNavier-
Stokes equations, defined in a domain which includes the symmetry axis, have
been obtained in [6].) We note that Theorem 3.3 leaves open the possibility that
there might exist Leray-Hopf weak solutions of the (3D) Navier-Stokes equations
with L2 axi-symmetric initial velocity, for which the symmetry is spontaneously
broken.
Remark3.3. Wehaveconsidered, throughout thissection, viscousflowswithzero
forcing. Itshould benoted that, iftheforcing term f doesnotvanish and respects
thesamesymmetryastheinitialvelocity,thentheproofsofTheorems3.1,3.2and
3.3canbeeasilyadapted toshowthat
t 2
kv−uk2 (t) ≤ kv −u k2 exp M ku k2 +2 hu(s,·),f(s,·)ids ,
L2(C) 0 0 L2(C) 0 L2(C)
( (cid:18) Z0 (cid:19) )
for some M = M(D,ν) > 0. This implies, clearly, continuous dependence with
respecttoinitialdataand,inparticular, uniqueness.
4. INVISCID FLOW
Inthissection wediscussthepossibility ofspontaneous symmetrybreaking for
the Euler system. Our first observation is that spontaneous symmetry breaking is
possibleforweaksolutionsoftheEulersystem,incontrastwithwhatweobserved
for the Navier-Stokes equations. This is a special case of a construction due to
De Lellis and Sze´kelyhidi in [1], see Proposition 2. We will use this construction
as formulated in Theorem 2 of [21]. Before we begin we need to introduce some
terminology. Since, in this section, we deal only with flows in a periodic box we
introduce thenotationQN = [0,1]N fortheperiodic boxinRN.
Definition 4.1. Let f ∈ L1(Q3). We say that f is essentially independent of x
3
(which is shortened to ei-x ) if, for almost every a,b ∈ (0,1), f(x ,x ,a) =
3 1 2
f(x ,x ,b),foralmostall(x ,x )∈ Q2.
1 2 1 2
Withthis,wearenowreadytostateprecisely thesymmetrybreaking result.
Theorem 4.1. Let u = (u1,u2) ∈ C∞(Q2) be divergence-free and periodic.
0 0 0
Thereexistsaweaksolution(infactinfinitelymany)u= u(t,x ,x ,x )∈ C0([0,∞);H (Q3))
1 2 3 w
of the incompressible 3D Euler equations such that u(t = 0) = (u ,0), and u is
0
notei-x .
3