Kinetic methods for Line–energy Ginzburg–Landau models
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Kinetic methods for Line–energy Ginzburg–Landau models Pierre-Emmanuel Jabin and Benoıt Perthame Departement de Mathematiques et Applications, UMR8553, Ecole Normale Superieure, 45, rue d'Ulm, 75230 Paris Cedex 05, France July 17, 2002 Abstract. A class of variational problems arising in thin micromagnetic film or in the gradient theory of phase transitions exhibit an hyperbolic behavior, a surprising property being given their natural elliptic structure. These two–dimensional Ginzburg–Landau problems are, for in- stance, characterized by energy density concentrations on a one–dimensional set - comparable to a steady shock wave. Here we review how methods based on kinetic formulations can help to understand some feautures of this broad and fascinating class of problems. Especially we deduce a general regularity result and also we characterize the zero-energy states and the domains where they can occur. Key words. Ginzburg–Landau energy, vortices, kinetic formulation, averaging lemmas, Sobolev spaces. AMS Class. Numbers. 35B65, 35J60, 35L65, 74G65, 82D30. Contents 1 Typical examples 1 2 kinetic formulation 3 3 A generic regularity result 4 4 Vortices and zero energy states 6 1 Typical examples Among the wide subject of Ginzburg-Landau variational problems, a typical problem is to study the limit as the parameter ? vanishes, for divergence free functions in R2 with a finite Ginzburg- Landau energy.
- thus does
- energy states
- averaging lemmas
- ginzburg–landau problems
- line–energy ginzburg–landau
- vlasov equation
- kinetic formulation
- ginzburg-landau variational
- scalar conservation
Publié par
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English
KineticmethodsforLine–energyGinzburg–Landaumodels
Pierre-EmmanuelJabinandBenoˆtPerthame
DepartementdeMathematiques
etApplications,UMR8553,
EcoleNormaleSuperieure,45,rued’Ulm,
75230ParisCedex05,France
July17,2002
Abstract.
Aclassofvariationalproblemsarisinginthinmicromagneticlmorinthegradient
theoryofphasetransitionsexhibitanhyperbolicbehavior,asurprisingpropertybeinggiven
theirnaturalellipticstructure.Thesetwo–dimensionalGinzburg–Landauproblemsare,forin-
stance,characterizedbyenergydensityconcentrationsonaone–dimensionalset-comparable
toasteadyshockwave.Herewereviewhowmethodsbasedonkineticformulationscanhelpto
understandsomefeauturesofthisbroadandfascinatingclassofproblems.Especiallywededuce
ageneralregularityresultandalsowecharacterizethezero-energystatesandthedomainswhere
theycanoccur.
Keywords.
Ginzburg–Landauenergy,vortices,kineticformulation,averaginglemmas,Sobolev
spaces.
AMSClass.Numbers.
35B65,35J60,35L65,74G65,82D30.
Contents
1Typicalexamples
2kineticformulation
3Agenericregularityresult
4Vorticesandzeroenergystates
1346
1Typicalexamples
AmongthewidesubjectofGinzburg-Landauvariationalproblems,atypicalproblemistostudy
thelimitastheparameter
ε
vanishes,fordivergencefreefunctionsin
R
2
withaniteGinzburg-
Landauenergy.Namely,weconsiderfunctions
u
ε
:
R
2
→
R
2
(
asmoothdomainof
R
2
)
suchthat
1
div
u
ε
=0in
,u
ε
n
=0on
∂
,
(1.1)
where
n
denotestheouterunitnormaltotheboundary
∂
of
R
2
,and,inthis
weakly
constrained
casetheenergyis
E
ε
(
u
ε
)=
ε
|r
u
ε
|
2
+1(1
|
u
ε
|
2
)
2
.
(1.2)
ZZε
Roughlyspeaking,thesemodelsintroducedinJin&Kohn[17],Ambrosio,DeLellis&Man-
tegazza[4],DeSimone,Kohn,Muller&Otto[9])comethroughdimensionalreductionofathree
dimensionalGinzburg–Landau–typemodelinathinlmandsingularlydependonthesmall
parameter
ε
proportionaltothelmthickness.Theyariseinmanyphysicalsituationslike
smecticliquidcrystals,softferromagneticlms,inblisterformationor—moreabstractly—in
thegradienttheoryofphasetransition(see[10]andthereferencestherein).Duetothevariety
ofthesesituations,tothecomplexityofthereductionandthenecessityofmathematicalsimpli-
cations,severalothermodelsareofinterest.ForinstanceRiviere&Serfaty[23]considerthe
stronglyconstrained
casewheretheconstraintisgivenby
|
u
|
=1in
,
(1.3)
whereasthefunctionalis
E
ε
2
(
u
)=
ε
|r
u
|
2
+1
|r
1
div
u
|
2
,
(1.4)
ZZε2R
whereforthelastterm,
u
hasbeentrivially(thatis,byzero)extendedonall
R
2
.Morerecently
amodelretainingthethreedimensionalaspectoftheseproblemshasbeenstudiedbyAllouges,
Riviere&Serfaty[2].
Manyanalyticalmethodsarisetostudythesekindsofproblems.Especially,variationalanaly-
sisarisesnaturally,
SBV
-typespaces,geometricalmeasuretheory
...
Thesereectstheelliptic-
ityoftheproblem.Moresurprisinglyentropies,compensatedcompactness,kineticformulations
andaveraginglemmasalsohelptoprovidepiecesofinformations.Thesereectsthehyperbolic
featureofthelimit
ε
→
0.
Inthispaperwewishtoillustrateaparticularpointwhichiswhy
kineticformulations
arise
andwhatkindofinformationitcanprovide.Letusinsistthatalternativetoolshavealso
beenusedintheabovementionedpapersandmostoftheresultsexplainedherecanbederived
dierently.Itseemshoweverthatthekineticstruturewhichariseshereisfascinatingenoughto
havealookatit,andespeciallytofurtherinvestigateinthesetermsthedierencebetweenthe
twomodelsmentionedaboveandalsotheirvariants.
Beforedoingthat,wewouldlikehowevertospendsometimetoexplainthehyperbolicaspect
inthelimit
ε
→
0intheseproblemsfollowing[9].Thisformallyappearsverynaturallybecause
inthislimitweobtain
|
u
|
=1
,
div
u
=0
,
2
)5.1(
in
.Intwodimensionsthisistypicallyascalarconservationlawwhichcanbealsoseenas
∂
cos(
)
∂
sin(
)
.0=+∂x
1
∂x
2
Thequestionistoprove(orasconjecturedfor(1.1),(1.2),rathertodisprove)thatitisthe
entropysolution.Anotherrelatedpointofviewistotransformtheproblemin
u
=
r
T
(
x
)
,
|r
|
=1
.
Thenthequestionistoknowweither(orratherwhynotinthecaseof(1.1),(1.2))
isthe
viscositysolutiontotheaboveeikonalequation.Alsonoticethatthisone-dimensionalaspect
ofthesingularsetmakesthisproblemverydierentformtheusualtwodimensionalGinzburg-
Landauproblem[6]wherepointvorticesappear,orofthethreedimensionalvortextubesasin
AftalionandJerrard[1].
Thepaperisorganizedasfollows.Thesecondsectionexplainswhyakineticformulationis
naturalhere,thethirdsectionshowstheregularitythatcanbededucedfromaveraginglemmas,
thelastsectionisdevotedtothecharacterizationofzeroenergystates.
2kineticformulation
Thekineticformulationarisesnaturallyinthelimitas
ε
vanishesintheproblem(1.1),(1.2).
Theoriginalmotivationcomesfromanargumentdevelopedin[9]andwhichisbasedona
familyofentropiesadaptedtothelimitaswrittenin(1.5)(dierententropiesarealsobuiltin
[17]whichproducesomewhatdierentproperties).And,thekineticformulationsintroducedin
Lions,PerthameandTadmor[19],[20]aimexactlytorepresentafullfamilyofentropiesbya
singlegenerating‘equilibrium’function,denotedby
below,bymeansofintegrationofanextra
variable.Thishastheadvantagetoreplaceaninnitefamilyofinequalitiesbyasingleequation
inahigherdimensionspace.Theextravariable,denotedby
inthispaper,ishomogeneoustoa
velocityandisthuscalledthekineticvariable.InthecontextofLine–energyGinzburg–Landau
modelsitcanbeintroduceddrectlythroughasimpleandgenerallemmawhichproofcanbe
foundinJabinandPerthame[14].
Lemma2.1
Foranysmoothfunction
u
de
