On the uniqueness of the solution of the two dimensional Navier Stokes equation with a Dirac mass as initial vorticity

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On the uniqueness of the solution of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity Isabelle Gallagher Universite de Paris 7 Institut de Mathematiques de Jussieu Case 7012, 2 place Jussieu 75251 Paris Cedex 05 France Thierry Gallay Institut Fourier Universite de Grenoble I 38402 Saint-Martin-d'Heres, France Pierre-Louis Lions CEREMADE Universite de Paris-Dauphine 75775 Paris cedex 16, France Abstract We propose two different proofs of the fact that Oseen's vortex is the unique solution of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity. The first argument, due to C.E. Wayne and the second author, is based on an entropy estimate for the vorticity equation in self-similar variables. The second proof is new and relies on symmetrization techniques for parabolic equations. 1 Introduction We consider the vorticity equation associated to the two-dimensional Navier-Stokes equation, namely ∂t?(x, t) + u(x, t) · ??(x, t) = ∆?(x, t) , x ? R2 , t > 0 . (1.1) The velocity field u(x, t) ? R2 is obtained from the vorticity ?(x, t) ? R via the Biot-Savart law u(x, t) = 12pi ∫ R2 (x? y)? |x? y|2 ?(y, t) dy , x ? R 2 , t > 0 , (1.2)

  • convection-diffusion equation

  • csiszar-kullback inequality

  • similar solution

  • vortex filament

  • using entropy estimates

  • self-similar variables

  • c1 function

  • oseen's vortex

  • biot-savart law


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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