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Navier–StokesEquationsandTurbulence
Thisbookaimstobridgethegapbetweenpracticingmathematiciansandthepractitionersof
turbulencetheory. Itpresentsthemathematicaltheoryofturbulencetoengineersandphysi-
cistsaswellasthephysicaltheoryofturbulencetomathematicians. Thebookistheresult
ofmanyyearsofresearchbytheauthors,whoanalyzeturbulenceusingSobolevspacesand
functionalanalysis. Inthiswaytheauthorshaverecoveredpartsoftheconventionaltheory
ofturbulence,derivingrigorouslyfromtheNavier–Stokesequationswhathadbeenarrivedat
earlierbyphenomenologicalarguments.
Themathematicaltechnicalitiesarekepttoaminimumwithinthebook,enablingthedis-
cussiontobeunderstoodbyabroadaudience. Eachchapterisaccompaniedbyappendices
thatgivefulldetailsofthemathematicalproofsandsubtleties.Thisuniquepresentationshould
ensureavolumeofinteresttomathematicians,engineers,andphysicists.
CiprianFoiasisanEmeritusProfessorintheDepartmentofMathematicsatIndianaUniversity
atBloomingtonandProfessorofMathematicsatTexasA&MUniversityatCollegeStation.He
hasheldnumerousvisitingprofessorships,includingthoseatVirjeUniversity(Netherlands),
IsraelInstituteofTechnology,UniversityofCaliforniaatSanDiego,UniversitéParis-Sud,and
theCollègedeFrance. In1995,hewasawardedtheNorbertWienerprizebytheAmerican
MathematicalSociety.
OscarP.Manleyworksasaconsultantandindependentresearcheronthefoundationsoftur-
bulentflows. HehasactedasheadoftheU.S.DepartmentofEnergy’sEngineeringResearch
ProgramandastheProgramManagerfortheDepartmentofEnergy’sresearchonmagnetic
fusion theory. Dr. Manley has held visiting professorships at the Université Paris-Sud and
IndianaUniversity.
RicardoRosaisaProfessorofMathematicsattheUniversidadeFederaldoRiodeJaneiro. He
hasalsoheldpositionsasVisitingResearcherattheInstituteforScientificComputingandAp-
pliedMathematicsatIndianaUniversityandasVisitingProfessorattheUniversitéParis-Sud
inOrsay,France.
RogerTemamisaProfessorofMathematicsattheUniversitéParis-SudandSeniorScientistat
theInstituteforScientificComputingandAppliedMathematicsatIndianaUniversity. Hehas
beenawardedanHonoraryProfessorshipatFudanUniversity(Shanghai),theFrenchAcad-
emyofScience’sGrandPrixAlexandreJoannidès,andtheSeymourCrayPrizeinNumerical
Simulation. ProfessorTemam has authored or co-authored nine books and published more
than260articlesininternationalrefereedjournals. Hiscurrentresearchinterestsinfluidme-
chanicsareintheareasofcontrolofturbulence,boundarylayertheory,andgeophysicalfluid
dynamics.
ENCYCLOPEDIAOFMATHEMATICSANDITSAPPLICATIONS
EDITEDBYG.-C.ROTA
EditorialBoard
R.Doran,P.Flajolet,M.Ismail,T.-Y.Lam,E.Lutwak
Volume83
Navier–StokesEquationsandTurbulence
6 H.Minc Permanents
19 G.G.Lorentz,K.Jetter,andS.D.Riemenschneider BirkhoffInterpolation
22 J.R.Bastida FieldExtensionsandGaloisTheory
23 J.R.Cannon TheOne-DimensionalHeatEquation
24 S.Wagon TheBanach–TarskiParadox
25 A.Salomaa ComputationandAutomata
27 N.H.Bingham,C.M.Goldie,andJ.L.Teugels RegularVariation
28 P.P.PetrushevandV.A.Popov RationalApproximationof RealFunctions
29 N.White(Ed.) CombinatorialGeometries
30 M.PohstandH.Zassenhaus AlgorithmicAlgebraicNumberTheory
31 J.AczelandJ.Dhombres FunctionalEquationsinSeveralVariables
32 M.Kuczma,B.Choczewski,andR.Ger IterativeFunctionalEquations
33 R.V.Ambartzumian FactorizationCalculusandGeometricProbability
34 G.Gripenberg,S.-O.Londen,andO.Staffans VolterraIntegralandFunctionalEquations
35 G.GasperandM.Rahman BasicHypergeometricSeries
36 E.Torgersen ComparisonofStatisticalExperiments
38 N.Korneichuk ExactConstantsinApproximationTheory
39 R.BrualdiandH.Ryser CombinatorialMatrixTheory
40 N.White(Ed.) MatroidApplications
41 S.Sakai OperatorAlgebrasinDynamicalSystems
42 W.Hodges BasicModelTheory
43 H.StahlandV.Totik GeneralOrthogonalPolynomials
45 G.DaPratoandJ.Zabczyk StochasticEquationsinInfiniteDimensions
46 A.Björneretal. OrientedMatroids
47 G.EdgarandL.Sucheston StoppingTimesandDirectedProcesses
48 C.Sims ComputationwithFinitelyPresentedGroups
49 T.Palmer BanachAlgebrasandtheGeneralTheoryof*-Algebras
50 F.Borceux HandbookofCategoricalAlgebra1
51 F.Borceux HandbookofCategoricalAlgebra2
52 F.Borceux HandbookofCategoricalAlgebra3
53 V.F.Kolchin RandomGraphs
54 A.KatokandB.Hasselblatt IntroductiontotheModernTheoryofDynamicalSystems
55 V.N.Sachkov CombinatorialMethodsinDiscreteMathematics
56 V.N.Sachkov ProbabilisticMethodsinDiscreteMathematics
57 P.M.Cohn SkewFields
58 R.Gardner GeometricTopography
59 G.A.Baker,Jr.andP.Graves-Morris PadéApproximants
60 J.Krajicek BoundedArithmeticPropositionalLogicandComplexityTheory
61 H.Groemer GeometricApplicationsofFourierSeriesandSphericalHarmonics
62 H.O.Fattorini InfiniteDimensionalOptimizationandControlTheory
63 A.C.Thompson MinkowskiGeometry
64 R.B.BapatandT.E.S.Raghavan NonnegativeMatriceswithApplications
65 K.Engel SpernerTheory
66 D.Cvetkovic,P.Rowlinson,andS.Simic EigenspacesofGraphs
67 F.Bergeron,G.Labelle,andP.Leroux CombinatorialSpeciesandTree-LikeStructures
68 R.GoodmanandN.R.Wallach RepresentationsandInvariantsoftheClassicalGroups
69 T.Beth,D.Jungnickel,andH.Lenz DesignTheory,vol.1
70 A.PietschandJ.Wenzel OrthonormalSystemsforBanachSpaceGeometry
71 G.E.Andrews,R.Askey,andR.Roy SpecialFunctions
72 R.Ticciati QuantumFieldTheoryforMathematicians
73 M.Stern SemimodularLattices
74 I.LasieckaandR.Triggiani ControlTheoryforPartialDifferentialEquationsI
75 I.LasieckaandR.Triggiani ControlTheoryforPartialDifferentialEquationsII
76 A.A.Ivanov GeometryofSporadicGroups1
77 A.Schinzel PolynomialswithSpecialRegardtoReducibility
78 H.Lenz,T.Beth,andD.Jungnickel DesignTheory,vol.2
encyclopedia of mathematics and its applications
Navier–Stokes Equations and Turbulence
C.FOIAS O.MANLEY
R.ROSA R.TEMAM
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
The Edinburgh Building, Cambridge CB2 2RU, UK
40 West 20th Street, New York, NY 10011-4211, USA
477 Williamstown Road, Port Melbourne, VIC 3207, Australia
Ruiz de Alarcón 13, 28014 Madrid, Spain
Dock House, The Waterfront, Cape Town 8001, South Africa
http://www.cambridge.org
©Cambridge University Press 2004
First published in printed format 2001
ISBN 0-511-03936-0 eBook (netLibrary)
ISBN 0-521-36032-3 hardback
Contents
Preface pageix
Acknowledgments xiv
ChapterI IntroductionandOverviewofTurbulence 1
Introduction 1
1. ViscousFluids.TheNavier–StokesEquations 1
2. Turbulence:WheretheInterestsof EngineersandMathematicians
Overlap 5
3. ElementsoftheTheoriesofTurbulenceof Kolmogorovand
Kraichnan 9
4. FunctionSpaces,FunctionalInequalities,andDimensionalAnalysis 14
ChapterII ElementsoftheMathematicalTheoryofthe
Navier–StokesEquations 25
Introduction 25
1. EnergyandEnstrophy 27
2. BoundaryValueProblems 29
3. Helmholtz–LerayDecompositionofVectorFields 36
4. WeakFormulationoftheNavier–StokesEquations 39
5. FunctionSpaces 41
6. TheStokesOperator 49
7. ExistenceandUniquenessofSolutions:TheMainResults 55
8. AnalyticityinTime 62
9. GevreyClassRegularityandtheDecayoftheFourierCoefficients 67
10. FunctionSpacesfortheWhole-SpaceCase 75
11. TheNo-SlipCasewithMovingBoundaries 77
12. DissipationRateofFlows 80
13. NondimensionalEstimatesandtheGrashofNumber 87
AppendixA. MathematicalComplements 90
AppendixB. ProofsofTechnicalResultsinChapterII 102
vii
viii Contents
ChapterIII FiniteDimensionalityof Flows 115
Introduction 115
1. DeterminingModes 123
2. DeterminingNodes 131
3. AttractorsandTheirFractalDimension 137
4. ApproximateInertialManifolds 150
AppendixA. ProofsofTechnicalResultsinChapterIII 156
ChapterIV StationaryStatisticalSolutionsoftheNavier–Stokes
Equations,TimeAverages,andAttractors 169
Introduction 169
1. MathematicalFramework,DefinitionofStationaryStatistical
Solutions,andBanachGeneralizedLimits 172
2. InvariantMeasuresandStationaryStatisticalSolutionsin
Dimension2 183
3. StationaryStatisticalSolutionsinDimension3 189
4. AttractorsandStationaryStatisticalSolutions 194
5. AverageTransferof EnergyandtheCascadesinTurbulentFlows 198
AppendixA. NewConceptsandResultsUsedinChapterIV 218
AppendixB. ProofsofTechnicalResultsinChapterIV 227
AppendixC. AMathematicalComplement:TheAccretivityProperty
inDimension3 244
ChapterV Time-DependentStatisticalSolutionsofthe
Navier–StokesEquationsandFullyDeveloped
Turbulence 255
Introduction 255
1. Time-DependentStatisticalSolutionsonBoundedDomains 262
2. HomogeneousStatisticalSolutions 271
3. ReynoldsEquationfortheAverageFlow 280
4. Self-SimilarHomogeneousStatisticalSolutions 283
5. RelationwithandApplicationtotheConventionalTheoryof
Turbulence 295
6. SomeConcludingRemarks 310
AppendixA. ProofsofTechnicalResultsinChapterV 312
References 331
Index 343
