LARGE TIME BEHAVIOR FOR VORTEX EVOLUTION IN THE HALF PLANE

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LARGE TIME BEHAVIOR FOR VORTEX EVOLUTION IN THE HALF-PLANE D. Iftimie M.C. Lopes Filho1 H.J. Nussenzveig Lopes2 Abstract. In this article we study the long-time behavior of incompressible ideal flow in a half plane from the point of view of vortex scattering. Our main result is that certain asymptotic states for half-plane vortex dynamics decompose naturally into a nonlinear superposition of soliton-like states. Our approach is to combine techniques developed in the study of vortex confinement with weak convergence tools in order to study the asymptotic behavior of a self-similar rescaling of a solution of the incompressible 2D Euler equations on a half plane with compactly supported, nonnegative initial vorticity. Key words: Incompressible and ideal fluid flow, vortex dynamics, 2D Euler equations. AMS subject classification: 76B47, 76B15, 35Q35. Contents 1. Introduction 1 2. Confinement of vorticity 5 2.1. Preliminary results 5 2.2. One-sided horizontal confinement of vorticity 6 2.3. Proximity to the boundary 9 2.4. Vertical confinement 11 3. Asymptotic behavior of nonnegative vorticity in the half-plane 11 3.1. Rescaled vorticity and asymptotic densities 12 3.2. The key estimate 13 3.3. Large time asymptotics 20 4. Extensions and Conclusions 24 Appendix. Separation of two vortices above a flat wall 25 References 28 1.

  • dynamics decompose naturally

  • ?˜ has bounded

  • dimensional dirac

  • vortex dynamics

  • naturally associated

  • vorticity

  • self-canceling when

  • vortex pairs

  • initial vorticity



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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