Maximal regularity in weighted spaces, nonlinear boundary conditions, and global attractors [Elektronische Ressource] / von Martin Meyries

Description

Maximal Regularity in Weighted Spaces,Nonlinear Boundary Conditions,and Global AttractorsZur Erlangung des akademischen Grades einesDOKTORS DER NATURWISSENSCHAFTENvon der Fakultät für Mathematik desKarlsruher Instituts für TechnologiegenehmigteDISSERTATIONvonMartin Meyriesaus Germersheim am RheinTag der mündlichen Prüfung: 24. November 2010Referent: Prof. Dr. Roland SchnaubeltKorreferenten: Prof. Dr. Jan Prüss, Prof. Dr. Lutz WeisFür meine ElternContentsIntroduction 11 The Spaces L and Weighted Anisotropic Spaces 11p; 1.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Abstract Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.2.1 Abstract Maximal L -Regularity . . . . . . . . . . . . . . . . . . . 28p; 1.2.2 Operator-Valued Fourier Multipliers . . . . . . . . . . . . . . . . . . 311.3 Weighted Anisotropic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 321.3.1 The Newton Polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.3.2 Temporal Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.3.3 Spatial Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.3.4 Pointwise Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 Maximal L -Regularity for Static Boundary Conditions 65p; 2.1 The Problem and the Approach in Weighted Spaces . . . . . . . . . . . . . 65n n2.

Sujets

• Mathematics

Informations

• Publié par: karlsruher_institut_fur_technologie
• Publié le: 01 janvier 2010
• Nombre de lectures: 36
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

MaximalRegularityinWeightedSpaces,
Conditions,BoundaryNonlinearttractorsAGlobaland

ZurErlangungdesakademischenGradeseines

WISSENSCHAFTENTURNADERDOKTORS

vonderFakultätfürMathematikdes
KarlsruherInstitutsfürTechnologie
genehmigte

TIONATDISSER

onv

MartinMeyries

RheinamGermersheimaus

TagdermündlichenPrüfung:24.November2010

Referent:Prof.Dr.RolandSchnaubelt

Korreferenten:Prof.Dr.JanPrüss,Prof.Dr.LutzWeis

Für

meine

Eltern

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1

1TheSpacesLandWeightedAnisotropicSpaces11
p,µ1.1BasicProperties.................................11
1.2AbstractProperties................................28
1.2.1AbstractMaximalL-Regularity...................28
p,µ1.2.2Operator-ValuedFourierMultipliers..................31
1.3WeightedAnisotropicSpaces..........................32
1.3.1TheNewtonPolygon...........................34
1.3.2TemporalTraces.............................38
1.3.3SpatialTraces...............................44
1.3.4PointwiseMultipliers...........................50

2MaximalLp,µ-RegularityforStaticBoundaryConditions65
2.1TheProblemandtheApproachinWeightedSpaces.............65
nn2.2TopOrderConstantCoefficientOperatorsonRandR..........72
+2.2.1TheFull-SpaceCasewithoutBoundaryConditions..........72
2.2.2TheHalf-SpaceCasewithBoundaryConditions............73
2.3TopOrderCoefficientshavingSmallOscillation................79
2.4TheGeneralCaseonaDomain.........................86
2.5ARight-InversefortheBoundaryOperator..................98

3MaximalL-RegularityforBoundaryConditionsofRelaxationType113
p,µ3.1TheProblemandtheApproachinWeightedSpaces.............113
3.2Half-SpaceProblemswithBoundaryConditions................125
3.2.1ConstantCoefficients...........................125
3.2.2TopOrderCoefficientshavingSmallOscillation............134
3.3TheGeneralCaseonaDomain.........................138

4AttractorsinStrongerNormsforRobinBoundaryConditions143
4.1Introduction....................................143
4.2SuperpositionOperators.............................146
4.3TheLocalSemiflow................................153
4.4GlobalAttractorsinStrongerNorms......................160

4.5Applications....................................164
4.5.1Reaction-DiffusionSystemswithNonlinearBoundaryConditions..164
4.5.2AChemotaxisModelwithVolume-FillingEffect...........165
4.5.3APopulationModelwithCross-Diffusion...............166

5BoundaryConditionsofReactive-Diffusive-ConvectiveType167
5.1Introduction....................................167
5.2MaximalL-RegularityfortheLinearizedProblem.............169
p,µ5.3TheLocalSemiflowforQuasilinearProblems.................173
5.4APrioriHölderBoundsimplyGlobalExistence................175
5.5TheGlobalAttractorforSemilinearDissipativeSystems...........180

191endixAppAnA.1BoundariesofDomainsinR..........................191
A.2InterpolationTheory...............................192
A.3SectorialOperators................................194
A.4FunctionSpacesonDomainsandBoundaries.................201
A.5DifferentialOperatorsonaBoundary......................205
A.6Gagliardo-NirenbergInequalities........................207

yBibliograph

209

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Thesubjectofthisthesisisthemathematicalanalysisoflinearandquasilinearparabolic
problemswithinhomogeneousandnonlinearboundaryconditions.Weconsiderstatic
boundaryconditionsofDirichlet,NeumannorRobintype,andfurtherboundarycon-
ditionsofrelaxationtype,whichincludedynamiconesaswellasboundaryconditionsthat
ariseinthelinearizationoffreeboundaryproblems.

Evolutionequationsofthistypedescribeagreatvarietyofphysical,chemicalandbiological
phenomena,likereaction-diffusionprocesses,phasefieldmodels,chemotacticbehaviour,
populationdynamics,phasetransitionsandthebehaviouroftwophasefluids,forinstance.
Inmanycasesitisnecessarytoimposenonlinearboundaryconditionsintoareaction-
diffusionmodeltocapturethedynamicsofthephenomenonunderinvestigation.Inthe
contextoffreeboundaryproblemsnonlinearboundaryconditionsnaturallyariseaftera
transformationtoafixeddomain.

WefocusonmaximalregularityresultsinweightedLp-spacesforlinearnonautonomous
parabolicproblemswithinhomogeneousboundaryconditions.Comparedtotheapproach
withoutweights,weareabletoreducethenecessaryregularityoftheinitialvalues,to
incorporateaninherentsmoothingeffectintothesolutionsandtoavoidcompatibility
conditionsattheboundary.Thesepropertiesserveusasabasisforconstructingalocal
semiflowforthecorrespondingquasilinearproblemsinascaleofphasespaces,andforthe
investigationofthelong-timebehaviourofsolutionsintermsofglobalattractors.

Ourapproachtoquasilinearproblemsthusreliesonlinearizationandagoodunderstanding
ofthelinearproblem.ThisideagoesbackatleasttoKato[58],Sobolevskii[77]and
Solonnikov[79].InasemigroupcontextitwascarriedoutbyGrisvard[46],DaPrato&
Grisvard[22],Amann[3,4,5,6,7],DaPrato&Lunardi[23],Lunardi[67]andPrüss[70].
Semilinearproblemscanbetreatedintheframeworkofanalyticsemigroups,seeHenry’s
[51].monograph

Maximalregularitymeansthatthereisanisomorphismbetweenthedataandthesolution
ofthelinearprobleminsuitablefunctionspaces.Havingestablishedsuchasharpregularity
resultforthelinearization,thecorrespondingquasilinearproblemcanbetreatedbyquite
simpletools,likethecontractionprincipleandtheimplicitfunctiontheorem.Thereare
approachesinspacesofcontinuousfunctions(seeAngenent[12]andClément&Simonett
[19]),inHölderspaces(seeLunardi[67])andinLp-spacesforp∈(1,∞)(seeClément

2

ionuctdIntro

&Li[17]andPrüss[70]).Formoredetailsandotherapproachestoquasilinearparabolic
problemswerefertothediscussionin[10].
Thethreementionedmaximalregularitysettingshaveadvantagesanddisadvantages.The
continuoussettingisquitesimple,butstrongrestrictionsontheunderlyingspacesare
necessary.IntheHöldersettingthenonlinearitiesareeasytohandleandtheapproach
isalsoapplicabletofullynonlinearproblems,butunpleasantcompatibilityconditionsat
theinitialtimearenecessaryandaprioriestimatesinhighnormsarerequiredtoshow
globalexistenceofsolutions.IntheLp-settingpowerfultoolsfromvector-valuedharmonic
analysisareavailable(andneeded!),butontheotherhandgeometricassumptionsonthe
underlyingspacesarerequiredandalsohereonehastoworkinhighnorms.Forafurther
discussionwereferagainto[10].InthisthesisweentirelyworkinanLp-framework.
TodecidewetheraconcretelinearproblemenjoysmaximalLp-regularityinasuitable
settingisnoteasy.Forlinearproblemswhichcanbereducedtoanabstractequationof
formthe∂tu(t)+Au(t)=f(t),t>0,u(0)=u0,(1)
onaBanachspaceE,whereAisaclosedanddenselydefinedoperatoronE,theoperator
summethod,asdevelopedbyDaPrato&Grisvard[21]andextendedbyDore&Venni[31]
andKalton&Weis[57],isappropriateinmanycases.Weis[85]characterizedthemaximal
Lp-regularitypropertiesofanoperatorintermsofR-sectoriality.IfEisaHilbertspace,
theneverynegativegeneratorofaboundedanalyticC0-semigroupenjoysmaximalLp-
regularity.Unfortunately,aHilbertspacesettingdoesoftennotseemtobesuitableforthe
applicationstoquasilinearproblems.
Totreatsecondorderparabolicdifferentialequationswithinhomogeneousornonlinear
boundaryconditionsinamaximalLp-regularityapproachonetypicallychoosesE=Lp,
E=Wp−1orEasaninterpolationspaceinbetweenasabasicunderlyingspace.IfEis
closetoWp−1thentheboundaryconditionsareapriorionlysatisfiedinaweaksense,butin
thiswaytheproblemcanbecastintheform(1)andoperatorsummethodsareavailable,
inprinciple.IfEisclosetoLp,thentheboundaryconditionscanbeunderstoodina
pointwisesense,butaformulationintheabstractform(1)doesnotseemtobepossiblein
areasonableway,ingeneral-thereisalwaysa’PDEpart’lefttodealwith.Anadvantage
ofchoosingEclosetoLpisthatgrowthconditionsonthenonlinearitiescanbeavoided.
Combiningoperatorsummethodswithtoolsfromvector-valuedharmonicanalysis,Denk,
Hieber&Prüss[24,25]andDenk,Prüss&Zacher[26]showedmaximalLp-regularitywith
Lpasanunderlyingspaceforalargeclassofvector-valuedparabolicproblemsofevenorder
withinhomogeneousboundaryconditions.In[25]problemswithboundaryconditionsof
statictypeareconsidered,i.e.,
∂tu+A(t,x,D)u=f(t,x),x∈Ω,t>0,
Bj(t,x,D)u=gj(t,x),x∈Γ,t>0,j=1,...,m,(2)
u(0,x)=u0(x),x∈Ω.

tioncduIntro

3

Thisincludesthelinearizationofreaction-diffusionsystemsandofphasefieldmodelswith
Dirichlet,NeumannandRobinconditions.In[26]theauthorsstudyproblemswithbound-
aryconditionsofrelaxationtype,i.e.,
∂tu+A(t,x,D)u=f(t,x),x∈Ω,t>0,
∂tρ+B0(t,x,D)u+C0(t,x,DΓ)ρ=g0(t,x),x∈Γ,t>0,
Bj(t,x,D)u+Cj(t,x,DΓ)ρ=gj(t,x),x∈Γ,t>0,j=1,...,m,(3)
u(0,x)=u0(x),x∈Ω,
ρ(0,x)=ρ0(x),x∈Γ,
whichincludesdynamicboundaryconditionsaswellasproblemsarisingaslinearizations
offreeboundaryproblemsthataretransformedtoafixeddomain.HereΩ⊂Rnisa
domainwithcompactsmoothboundaryΓ=∂Ω.Thecoefficientsoftheoperatorsareonly
assumedtobepointwisemultiplierstotheunderlyingspaces,andthetopordercoefficients
arerequiredtobeboundedanduniformlycontinuous.Theseregularityassumptionsallow
toapplythelinearresultstoquasilinearproblems.Earlierinvestigationson(2)startedat
leastwithLadyzhenskaya,Solonnikov&Ural’ceva[64]andincludealsoWeidemaier[84].
AprincipleshortcomingofthemaximalLp-regularityapproachto(1),(2)and(3)isthat
forfixedponecansolvetheequationforinitialvaluesonlyinonesinglespaceofrelatively
highregularity,andthatonedoesnothavetheflexibilitytoworkinascaleofsp