Differential Equations with singular fields

Pierre-Emmanuel Jabin

Description

Differential Equations with singular fields Pierre-Emmanuel Jabin email: Equipe Tosca, Inria, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis Laboratoire Dieudonne, Univ. de Nice, Parc Valrose, 06108 Nice cedex 02 Abstract. This paper investigates the well posedness of ordinary differ- ential equations and more precisely the existence (or uniqueness) of a flow through explicit compactness estimates. Instead of assuming a bounded di- vergence condition on the vector field, a compressibility condition on the flow (bounded jacobian) is considered. The main result provides existence under the condition that the vector field belongs to BV in dimension 2 and SBV in higher dimensions. 1 Introduction This article studies the existence (and secondary uniqueness) of a flow for the equation ∂tX(t, x) = b(X(t, x)), X(0, x) = x. (1.1) The most direct way to establish the existence of such of flow is of course through a simple approximation procedure. That means taking a regularized sequence bn ? b, which enables to solve ∂tXn(t, x) = bn(X(t, x)), Xn(0, x) = x, (1.2) by the usual Cauchy-Lipschitz Theorem. To pass to the limit in (1.2) and obtain (1.1), it is enough to have compactness in some strong sense (in L1loc for instance) for the sequence Xn.

• then xn


• divergence condition instead


• let xn


• dimension


• kind can


• provides quantitative


• full bressan's


• right result

Sujets

• Antipolis
• Dimensión

Informations

• Publié par: pefav
• Nombre de lectures: 7
• Langue: English

Extrait

DifferentialEquationswithsingularfields

Pierre-EmmanuelJabin
email:[e-mail]
EquipeTosca,Inria,2004routedesLucioles,BP93,06902SophiaAntipolis
LaboratoireDieudonne´,Univ.deNice,ParcValrose,06108Nicecedex02

Abstract.
Thispaperinvestigatesthewellposednessofordinarydiffer-
entialequationsandmorepreciselytheexistence(oruniqueness)ofaflow
throughexplicitcompactnessestimates.Insteadofassumingaboundeddi-
vergenceconditiononthevectorfield,acompressibilityconditionontheflow
(boundedjacobian)isconsidered.Themainresultprovidesexistenceunder
theconditionthatthevectorfieldbelongsto
BV
indimension2and
SBV
inhigherdimensions.

1Introduction
Thisarticlestudiestheexistence(andsecondaryuniqueness)ofaflowfor
theequation
∂
t
X
(
t,x
)=
b
(
X
(
t,x
))
,X
(0
,x
)=
x.
(1.1)
Themostdirectwaytoestablishtheexistenceofsuchofflowisofcourse
throughasimpleapproximationprocedure.Thatmeanstakingaregularized
sequence
b
n
→
b
,whichenablestosolve

∂
t
X
n
(
t,x
)=
b
n
(
X
(
t,x
))
,X
n
(0
,x
)=
x,
(1.2)

bytheusualCauchy-LipschitzTheorem.Topasstothelimitin(1.2)and
obtain(1.1),itisenoughtohavecompactnessinsomestrongsense(in
L
l
1
oc
forinstance)forthesequence
X
n
.Obviouslysomeconditionsareneeded.
Firstnotethatweareinterestedinflows,whichmeansthatwearelooking
forsolutions
X
whichareinvertible:Atleast
JX
=det
d
x
X
6
=0
a.e
(with

1

d
x
X
thedifferentialof
X
in
x
only).Sothroughoutthispaper,onlyflows
x
→
X
(
t,x
)whicharenearlyincompressibleareconsidered
1≤
JX
(
t,x
)
≤
C,
∀
t
∈
[0
,T
]
,x
∈
R
d
.
(1.3)
CIfoneobtains
X
asalimitof
X
n
thenthemostsimplewayofsatisfying(1.3)
istohave
1≤
JX
n
(
t,x
)
≤
C,
∀
t
∈
[0
,T
]
,x
∈
R
d
,
(1.4)
Cforsomeconstant
C
independentof
n
.Notethatbothconditionsareonly
requiredonafiniteandgiventimeinterval[0
,T
]sinceonemayeasilyextend
X
over
R
+
bythesemi-grouprelation
X
(
t
+
T,x
)=
X
(
t,X
(
T,x
)).Usually
(1.3)and(1.4)areobtainedbyassumingaboundeddivergenceconditionon
b
or
b
n
butthisisnotthecasehere.
Itiscertainlydifficulttoguesswhatistheoptimalconditionon
b
.Itis
currentlythoughtthat
b
∈
BV
(
R
d
)isenoughor(see[12])
Bressan’scompactnessconjectur
R
e:
Let
X
n
beregular(
C
1
)solutionsto
(1.2)
,satisfying
(1.4)
andwith
sup
n
R
d
|
db
n
(
x
)
|
dx<
∞
.Thenthesequence
X
n
islocallycompactin
L
1
([0
,T
]
×
R
d
)
.
Fromthis,onewoulddirectlyobtaintheexistenceofaflowto(1.1)provided
that
b
∈
BV
(
R
d
)and(1.3)holds.InsteadofthefullBressan’sconjecture,
thisarticleessentiallyrecovers,throughadifferentmethod,theresultof[4]
namelyunderthecondition
b
∈
SBV
Theorem1.1
Assumethat
b
∈
SBV
loc
(
R
d
)
∩
L
∞
(
R
d
)
withalocallyfinite
jumpset(forthe
d
−
1
dimensionalHausdorffmeasure).Let
X
n
beregular
solutionsto
(1.2)
,satisfying
(1.4)
andsuchthat
b
n
→
b
belongsuniformlyto
L
∞
(
R
d
)
∩
W
1
,
1
(
R
d
)
.Then
X
n
islocallycompactin
L
1
([0
,T
]
×
R
d
)
.
Onlyindimension2isitpossibletobemoreprecise
Theorem1.2
Assumethat
d
≤
2
,
b
∈
BV
loc
(
R
d
)
.Let
X
n
beregularso-
lutionsto
(1.2)
,satisfying
(1.4)
andsuchthat
b
n
→
b
belongsuniformly
to
L
∞
(
R
d
)
∩
W
1
,
1
(
R
d
)
with
inf
K
b
n
∙
B>
0
foranycompact
K
andsome
B
∈
W
1
,
∞
(
R
d
,
R
d
)
.Then
X
n
islocallycompactin
L
1
([0
,T
]
×
R
d
)
.
Theproofofthefirstresultisfoundinsection7andtheproofofthesecond
insection6.Afternotationsandexamplesinsection2andtechnicallemmas
insection3,particularcasesarestudied.Insection4,averysimpleproof

2

isgivenif
b
∈
W
1
,
1
.Section5studies(1.1)indimension1forwhichcom-
pactnessholdsunderverygeneralconditions(essentiallynothingfor
b
and
amuchweakerversionof(1.3)).Thefinalsectionofferssomecommentson
theunresolvedissuesinthefull
BV
case.
Thequestionofuniquenessisdeeplyconnectedtotheexistenceandinfact
theproofofTheorem1.1maybeslightyalteredinordertoprovideit(itis
morecomplicatedforTh.1.2).Proofsarealwaysgivenforthecompactness
ofthesequencebutitisindicatedandbrieflyexplainedafterthestatedresults
whethertheycanalsogiveuniqueness;Thisisusuallythecaseexceptfor
sections5and6.
Thewellposednessof(1.1)isclassicallyobtainedbytheCauchy-Lipschitz
theorem.Thisisbasedonthesimpleestimate
|
X
(
t,x
+
δ
)
−
X
(
t,x
)
|≤|
δ
|
e
t
k
db
k
L
∞
.
(1.5)
Noticethatasimilarboundholdsif
b
isonlylog-lipschitz,leadingtothe
importantresultofuniquenessforthe2dincompressibleEulersystem(see
forinstance[28]amongmanyotherreferences).
Theideainthisarticleistoget(1.5)for
almostall
x
.Itisthereforegreatly
inspiredbytherecentapproachdevelopedin[18](seealso[17])wherethe
authorscontrolthefunctional
ZZsuplog1+
|
X
(
t,x
+
rw
)
−
X
(
t,x
)
|
dwdx.
(1.6)
R
d
rS
d
−
1
r
ThisallowsthemtogetanequivalentofTh.1.1provided
b
∈
W
1
,p
with
p>
1(andinfact
db
in
L
log
L
wouldwork).Heretheslightydifferent
functional
ZZsuplog1+
|
X
(
t,x
+
rw
)
−
X
(
t,x
)
|
dwdx,
(1.7)
r
R
d
S
d
−
1
r
isessentiallyconsidered.Itgivesabettercondition
b
∈
SBV
buthastheone
drawbackofnotimplyingstrongdifferentiabilityfortheflowatthelimit.
Othersuccessfulapproachesofcourseexistfor(1.1).Amostimportantstep
wasachievedin[22]wherethewellposednessoftheflowwasobtainedby
provinguniquenessfortheassociatedtransportequation
∂
t
u
+
b
∙r
u
=0
,
3

undertheconditionsthat
b
beofboundeddivergenceandin
W
1
,
1
.Thecru-
cialconceptthereistheoneofrenormalizedsolutions,namelyweaksolutions
u
s.t.
φ
(
u
)isalsoasolutiontothesametransportequation.Thiswasex-
tendedin[29],[27]and[26];alsosee[8]wheretheconnectionbetweenwell
posednessforthetransportequationand(1.1)isthoroughlyanalysed,both
forboundeddivergencefieldsandanequivalentto(1.3).
Usingaslightdifferentrenormalizationfortheequation
∂
t
u
+
r∙
(
bu
)=0,
thewellposednesswasfamouslyobtainedin[2]underthesamebounded
divergenceconditionand
b
∈
BV
.Thislastassumption
b
∈
BV
wasalso
consideredin[15]and[16].Stillusingrenormalizedsolutions,therestriction
ofboundeddivergencewasweakenedin[4]toanassumptionequivalentto
(1.3);Unfortunatelythisrequired
b
∈
SBV
.
Incomparisonto[4],Theorem1.1isslightyweaker(duetotheassumption
ofboundedjumpsetfor
H
d
−
1
),exceptin2
d
where(1.2)isstronger(
b
∈
BV
insteadof
SBV
).Themainadvantageoftheapproachpresentedhereisthat
weworkdirectlyonthedifferentialequation,givingforinstanceaverysimple
anddirecttheoryfor
b
∈
W
1
,
1
.Italsoprovidesquantitativeestimatesfor
|
X
(
t,x
)
−
X
(
t,x
+
δ
)
|
,whichisconnectedtotheregularityofthetrajectories.
Usuallyusingtransportequationsfor(1.1)doesnotdirectlygivesestimates
like(1.5),(1.6)or(1.7).Someinformationofthiskindcanstillbederi