LARGE TIME BEHAVIOR FOR VORTEX EVOLUTION IN THE HALF PLANE
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- dynamics decompose naturally
- ?˜ has bounded
- dimensional dirac
- vortex dynamics
- naturally associated
- vorticity
- self-canceling when
- vortex pairs
- initial vorticity
- ?˜
Publié par
Langue
English
Ontheuniquenessofthesolutionofthetwo-dimensional
Navier-StokesequationwithaDiracmassasinitialvorticity
IsabelleGallagherThierryGallay
UniversitedeParis7InstitutFourier
InstitutdeMathematiquesdeJussieuUniversitedeGrenobleI
Case7012,2placeJussieu38402Saint-Martin-d’Heres,France
75251ParisCedex05France
Pierre-LouisLions
CEREMADE
UniversitedeParis-Dauphine
75775Pariscedex16,France
Abstract
WeproposetwodierentproofsofthefactthatOseen’svortexistheuniquesolution
ofthetwo-dimensionalNavier-StokesequationwithaDiracmassasinitialvorticity.The
rstargument,duetoC.E.Wayneandthesecondauthor,isbasedonanentropyestimate
forthevorticityequationinself-similarvariables.Thesecondproofisnewandrelieson
symmetrizationtechniquesforparabolicequations.
1Introduction
Weconsiderthevorticityequationassociatedtothetwo-dimensionalNavier-Stokesequation,
namely
∂
t
ω
(
x,t
)+
u
(
x,t
)
r
ω
(
x,t
)=
ω
(
x,t
)
,x
∈
R
2
,t>
0
.
(1.1)
Thevelocityeld
u
(
x,t
)
∈
R
2
isobtainedfromthevorticity
ω
(
x,t
)
∈
R
viatheBiot-Savartlaw
21
Z
(
x
y
)
⊥
u
(
x,t
)=2
2
|
x
y
|
2
ω
(
y,t
)d
y,x
∈
R
,t>
0
,
(1.2)
Rwhere(
x
1
,x
2
)
⊥
=(
x
2
,x
1
).Itsatisesdiv
u
=0and
∂
1
u
2
∂
2
u
1
=
ω
.Equations(1.1),(1.2)
areinvariantunderthescalingtransformation
ω
(
x,t
)
7→
2
ω
(
x,
2
t
)
,u
(
x,t
)
7→
u
(
x,
2
t
)
,>
0
.
(1.3)
TheCauchyproblemforthevorticityequation(1.1)isgloballywell-posedinthe(scale
invariant)Lebesguespace
L
1
(
R
2
),seeforinstance[6].Toincludemoregeneralinitialdata,such
asisolatedvorticesorvortexlaments,itisnecessarytouselargerfunctionspaces.Anatural
candidateisthespace
M
(
R
2
)ofallniterealmeasureson
R
2
,equippedwiththetotalvariation
norm.Thisspacecontains
L
1
(
R
2
)asaclosedsubspace,anditsnormisinvariantunder(the
spatialpartof)therescaling(1
R
.3).Anotheru
R
sefultopologyon
M
(
R
2
)istheweakconvergence,
denedasfollows:
n
*
if
R
2
ϕ
d
n
→
R
2
ϕ
d
foranycontinuousfunction
ϕ
:
R
2
→
R
vanishingatinnity.
1
Existenceofsolutionsof(1.1)withinitialdatain
M
(
R
2
)wasrstprovedbyCottet[10],
andindependentlybyGiga,MiyakawaandOsada[13],seealsoKato[15].Uniquenesscan
beobtainedbyastandardGronwallargumentifthe
atomicpart
oftheinitialvorticity
is
sucientlysmall[13,15],butthismethodisboundtofailif
containslargeDiracmasses.In
theparticularcasewhere
=
0
forsome
∈
R
,anexplicitsolutionisknown:
x
x
ω
(
x,t
)=
G
√
,u
(
x,t
)=
√
v
G
√
,x
∈
R
2
,t>
0
,
(1.4)
tttterehw1
2
1
⊥
2
G
(
)=e
|
|
/
4
,v
G
(
)=1
e
|
|
/
4
,
∈
R
2
.
(1.5)
4
2
|
|
2
Thisself-similarsolutionofthetwo-dimensionalNavier-Stokesequationisoftencalledthe
Lamb-
Oseenvortex
withtotalcirculation
.Itistheuniquesolutionwithinitialvorticity
0
inthe
followingprecisesense:
Theorem1.1[12]
Let
T>
0
,
K>
0
,
∈
R
,andassumethat
ω
∈
C
0
((0
,T
)
,L
1
(
R
2
)
∩
L
∞
(
R
2
))
isasolutionof(1.1)satisfying
k
ω
(
,t
)
k
L
1
K
forall
t
∈
(0
,T
)
and
ω
(
,t
)
*
0
as
t
→
0+
.Then
x
ω
(
x,t
)=
G
√
,x
∈
R
2
,t
∈
(0
,T
)
.
ttHereandinthesequel,wesaythat
ω
∈
C
0
((0
,T
)
,L
1
(
R
2
)
∩
L
∞
(
R
2
))isa(mild)solution
of(1.1)iftheassociatedintegralequation
t2Zω
(
,t
2
)=e
(
t
2
t
1
)
ω
(
,t
1
)
r
e
(
t
2
t
)
u
(
,t
)
ω
(
,t
)d
t
(1.6)
t1issatis
