Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

3035 Pages·2018·39.187 MB·English

by  Yoshikazu Giga, Antonín Novotný

Yoshikazu Giga
Antonín Novotný
Editors
Handbook of
Mathematical
Analysis in
Mechanics of
Viscous Fluids
Handbook of Mathematical Analysis
in Mechanics of Viscous Fluids
Yoshikazu Giga • Antonín Novotný
Editors
Handbook of
Mathematical Analysis
in Mechanics of Viscous
Fluids
With62Figuresand3Tables
123
Editors
YoshikazuGiga AntonínNovotný
GraduateSchoolofMathematicalSciences UniversitédeToulon
UniversityofTokyo IMATH,Toulon
Meguro-ku,Tokyo France
Japan
ISBN978-3-319-13343-0 ISBN978-3-319-13344-7(eBook)
ISBN978-3-319-13345-4(printandelectronicbundle)
https://doi.org/10.1007/978-3-319-13344-7
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Preface
FluidmechanicshasalonghistorysinceGreekphilosopherArchimedesdiscovered
hisfamouslawofforcesactingonbodiesinmotionlessfluid.Agoodunderstanding
ofitsprinciplesanditsmathematicalformulationandpropertiesiscrucialinmany
branchesofcontemporaryscienceandtechnology.Notonlyitsmodernfoundation
relies on mathematics but also many branches of mathematics were developed or
evenemergedthroughtheresearchinfluidmechanicsorthroughthemathematical
formulation of problems of fluid mechanics. The examples include the theory
of functions of one complex variable, topology, dynamical systems, differential
equations,differentialgeometry,probabilitytheory,andfunctionalanalysis,toname
onlyafew.
Mathematics has been always playing a key role in the research on fluid
mechanics.Manyimminentproblemsinvariousbranchesofmathematicshavetheir
originorcanbeinterpretedasproblemsoffluidmechanicsalthoughinmanycases
thecommunityofpuremathematiciansfailstonoticethisfact.
The purpose of this handbook is to provide a synthetic review of the state
of the art in the theory of viscous fluids, present fundamental notions, formulate
problemsoffluidmechanicsrepresentingthedevelopmentofthetheoryduringlast
severaldecades,andshowthemethodsandmathematicaltoolsfortheirresolution.
Since the field of mathematical fluid mechanics is huge, it is impossible to cover
all topics. In this handbook, we focus on mathematical analysis in mechanics of
viscous Newtonian fluids. The first part consisting of two chapters is devoted to
derivation of basic equations by physical modeling. The second part is devoted to
mathematicalanalysisofincompressiblefluids,whilethethirdpartisdealingwith
themathematicaltheoryofviscouscompressiblefluids.Therearemanytopicsthat
arenotcoveredbythehandbook.Inparticular,thisisthecaseofnumericalanalysis
oftheequationswhichwoulddeservebyitselfanindependentvolume.
Thehandbookreviewsimportantproblemsandnotionsthatmarkedthedevelop-
mentofthetheory.Itexplainsthemethodsandtechniquesthatmaybeusedfortheir
resolution.Wehopethatitwillbeusefulnotonlytomathematicianswhoworkon
thefuturedevelopmentofthetheorybutalsotophysicistsandengineerswhoneed
toknowthetoolsofmathematicalanalysisfordevelopingapplications.
vii
viii Preface
Part1:DerivationofBasicEquations
There are several ways to derive the basic equations. One of the most typical
waysistostartfromthebalanceequations formass,momentum,andtotalenergy
incorporating to the process the second law of thermodynamics. This approach is
discussed in the first chapter by J. Málek and V. Pru˚ša. The approach based on
variationalprinciplesisdiscussedinthecontributionbyM.-H.Giga,A.Kirshtein,
andC. Liu.
Part2:IncompressibleFluids
TheNavier-Stokessystemisaconventionalmacroscopicmodelconsistingofpartial
differential equations describing the motion of viscous incompressible Newtonian
fluids.Themodernmathematicalanalysisoftheseequationsgoesbacktoseminal
worksofJ.Lerayin1933and1934.Lerayintroducedthenotionofweaksolutions
for the Navier-Stokes equations and developed several key tools of functional
analysisfortheirmathematicaltreatment.
Inhisabovementionedpapers,Lerayformulatedmanyimminentopenproblems.
One of them, on the regularity of weak solutions, was named among the seven
MillenniumPrizeProblemsoftheClayInstituteofMathematics.Thefundamental
questioniswhether,inthreespacedimensions,theglobalintime(weak)solutions
of the Navier-Stokes system emanating from the smooth initial data must be
smooth. This problem is related to the question whether or not the system of
Navier-Stokesequationsisstillagoodmodelforfluidflowsinregimeswithlarge
Reynoldsnumbers.Despitealotofeffortofexcellentmathematicians,thisproblem
is still open. Many mathematical tools and specific techniques, e.g., the theory
of partial differential equations, theory of interpolation, and maximal regularity
theory,havebeendevelopedorrefinedwithintheprocessofitsinvestigation.Many
important open problems have been formulated during its investigation, and some
ofthemsolved.
The accelerated development of technology at the end of the last century gave
rise to many new mathematical problems in fluid mechanics. We can name as
examples free boundary problems and problems related to complex fluids. In all
theseproblems,theNavier-Stokesequationsplayanimportantorevenacrucialrole.
Weshallincludeinthehandbookthesemoderntopics,aswellasrecentdevelopment
intheproblemsoninviscidlimits.
In this part, we intend to provide key notions and tools for mathematical
understanding of equations of incompressible fluids. We mainly discuss existence
and uniqueness problems as well as behavior of solutions and different notions of
theirstability.
When the flow is slow, or more precisely when the Reynolds number is small,
it is convenient to consider a linearized version of Navier-Stokes equations called
Stokes equations. The investigation of the Stokes system is not only extremely
importantforitself,butitisfundamentalalsoforthedevelopmentofthenonlinear
theory. Well-posedness and regularity questions for the various initial-boundary
value problems for the Stokes equations in various types of sufficiently smooth
domains are discussed in the first chapter by M. Hieber and J. Saal. Similar
Preface ix
problemsindomainswithlessregularboundariesareinvestigatedbyS.Monniaux
andZ.Shen.
There are several long-standing problems opened by Leray for the stationary
Navier-Stokes equations related to general inflow-outflow boundary conditions in
the case of bounded domains and to velocity profiles at infinity in the case of
unbounded domains. The recent development in the former case is reported in the
contributionbyM.V.Korovkov,K.Pileckas,andR.Russo,whilethecontribution
of T. Hishida deals with the latter case in 3-D exterior domains. The contribution
of G.P.Galdiand J.Neustupa isdevoted tothestationaryflows around arotating
body.
AftertheshortexcursiontothestationaryNavier-Stokesequations,thehandbook
continues by several chapters dealing with weak and strong solutions to the
nonsteadyNavier-Stokesequations.
The existence of a weak solution in a smooth general domain is discussed
by R. Farwig, H. Kozono, and H. Sohr. Self-similar solutions are introduced and
investigatedbyH.Jia,V.Šverák,andT.-P.Tsai.Theproblemofexistenceoftime
periodic solutions is addressed in the contribution by G. P. Galdi and M. Kyed.
LargetimebehaviorofsolutionsisdiscussedbyL.BrandoleseandM.E.Schonbek.
Since the Navier-Stokes equations have a regularizing property, it is interesting to
investigatethestructureofsetsof“rough”initialdatastillallowingsmoothsolutions
globally in time. These questions are addressed in the contribution by I. Gallager.
Stability of some special solutions, e.g., of the so-called Lamb-Oseen vortex, is
investigated by T. Gallay and Y. Maekawa. Some important exact solutions are
constructed in the contribution by H. Okamoto. Asymptotic behavior of solutions
neartheboundaryendowedwithno-slipconditionsisinvestigatedinthechapterby
Y.MaekawaandA. Mazzucato.Regularityandregularitycriteriaforweaksolutions
arediscussedbyG.SereginandV.ŠverákandbyH.BeirãodaVeiga,Y.Giga,and
Z.Grujicfromseveraldifferentviewpoints.Behaviorofsolutionofanidealflowis
discussedbyT.Y.HouandP.Liu.
The Navier-Stokes flow coupled with other effects is increasingly important.
A class of models for geophysical flows is discussed by J. Li and E. S. Titi.
EquationsforpolymetricmaterialsareinvestigatedbyN.Masmoudi,whilenematic
liquid crystal flows are discussed by M. Hieber and J. Prüss. Some problems for
viscoelasticfluidsarediscussedbyX.Hu,F.-H.Lin,andC.Liu.
A problem dealing with two-phase flows is a typical free boundary problem.
There are several chapters devoted to this topic by J. Prüss and S. Shimizu; by
V.A.SolonnikovandI.V.Denisova;byG.SimonettandM.Willa;andbyH.Abels
and H. Garke. The first three chapters handle smooth solutions, while the latter
chapterhandlesweaksolutionswhichallowtopologicalchangeofregionoccupied
by the fluid. Finally, the classical free boundary value problem of water wave is
discussedbyD.CórdovaandC.Fefferman.
Part3:CompressibleFluids
The mathematical models that take into account the compressibility of the fluid
andthermodynamicaleffectsleadtoarichvarietyofsystemsofpartialdifferential
x Preface
equations. Their mathematical character depends on the physical assumptions on
transportphenomena(describingtheviscouseffectsandtheheattransport)andon
constitutive laws (usually prescribing pressure and internal energy as functions of
densityandtemperature),notspeakingaboutthesituationswhentheseequationsare
coupledwithsystemsdescribingotherphenomena(e.g.,inmagnetohydrodynamics,
multifluid models, and theory of chemically reacting mixtures). In these circum-
stances, the classical formulation may give rise to different nonequivalent weak
formulations, which are all physically reasonable. This abundance of possibilities
imposes severe restrictions on the material that can be treated in one handbook
volume.
In particular, as in the “incompressible part” we have restricted our subject of
interest only to the so-called Newtonian fluids (when, loosely speaking, the stress
tensordependslinearlyonthegradientofvelocity),andtheheatfluxisproportional
to the gradient of temperature through the Fourier law. The resulting system
is called Navier-Stokes-Fourier system. This system describes the most simple
thermodynamically consistent model of fluid dynamics. If the pressure depends
on density only, one obtains a simpler model communally called compressible
Navier-Stokesequationsinbarotropicregime.Althoughsimplerthanthecomplete
Navier-Stokes-Fouriersystem,itinheritsmostofitsmathematicaldifficulties.The
presenthandbookisexclusivelytreatingthetwoabovementionedsystems.
The mathematical difficulties encountered in mathematical investigation of
compressible Navier-Stokes equations are many fold: they are related to (1) the
enormous range of scales of motion described by the system, (2) the absence of
mechanisms preventing density to create a vacuum and the absence of dissipation
inthemassconservation,(3)themixedparabolic-hyperbolic(orelliptic-hyperbolic
in the steady case) character of the underlying linearized equations, and (4) the
nonlinear character of equations (which still allows, through a priori estimates,
to prevent the field quantities of concentrations but does not defend them from
oscillations).
The history of mathematical investigation of the compressible Navier-Stokes
equations is more recent than the history of investigation of the incompressible
Navier-Stokes equations. It starts with the works of D. Graffi and J. Nash on
the local in time existence of strong solutions in the 1960s and continues with
global in time existence results for data close to the equilibrium in works of
A. Matsumura and T. Nishida in the 1980s. The first results on the existence of
weak solutions similar to those introduced for incompressible fluids by Leray in
1934wereestablishedbyP.-L.Lionsonlyin1998.Likewise,thestabilityanalysis
introduced for the Navier-Stokes equations by Prodi and Serrin in the 1960s was
waiting for its “compressible” counterpart till 2012. Being much younger than
its “incompressible” counterpart, the mathematical theory of compressible fluids
undergoes still a tumultuous development, and the synthetic level of contributions
in “compressible” part is therefore necessarily objectively less high than in the
“incompressible”part.
Thehandbook chaptersreportthestateoftheartofsomepartsofthetheoryat
the end of 2016. It covers mostly the questions of well-posedness of solutions to
Preface xi
variousboundaryandinitial-boundaryvalueproblemstotheseequations(meaning
the existence of strong and weak solutions, uniqueness or conditional uniqueness,
stability, longtime behavior, conditional regularity, and qualitative properties of
solutions) in one and several space dimensions. We have made a preference to
cover in the handbook solely these topics knowing well that this is a subjective
andnonexhaustingchoice.
Various types of weak and strong solutions are introduced and discussed in
the contribution of E. Feireisl. Qualitative theory of strong solutions has longer
history and its solid foundations are nowadays relatively well established. These
questions are treated in three chapters by R. Danchin; J. Burczak, Y. Shibata, and
W.M.Zaja¸czkowski;andM. Kotschote.Relatedquestionsofblowupcriteriaand
existenceofstrongsolutionsforsmalldatawithlargeoscillationsaredevelopedin
two contributions by Z. Huang and Z. Xin and by J. Li and Z. Xin. The longtime
behavior of strong solutions is investigated in the contribution of Y. Shibata and
Y.Enomoto,whileanoverviewofsomeresultsforfreeboundaryvalueproblemsis
giveninthecontributionbyI.DenisovaandV.A. Solonnikov.
The existence, stability, and asymptotic behavior of solutions to the equations
in one dimension, or of spherically and axially symmetric solutions, are treated in
threechaptersbyA.Zlotnik,byY.Qin,andbyS.JiangandQ.Ju.Thechapterby
A.Matsumuraisacomprehensiveintroductiontowavesin1-Dcompressiblefluids.
The theory of weak solutions has a short history and is still in agitated
development.Theexistenceofdifferenttypesofweaksolutions,stability,longtime
behavior,andweak-stronguniquenessprinciplearediscussedbyA.Novotnýandby
H.Petzeltová.Theexistenceproblemforweaksolutionswithdegenerateviscosity
coefficients is presented by D. Bresch and B. Desjardins. The contribution by
P. I. Plotnikov and W. Weigant is devoted to the existence results in the case of
critical adiabatic coefficients. A particular role in the theory of weak solutions is
playedbysolutionsbelongingtotheso-calledintermediateregularityclass.These
solutionsareintroducedinthecontributionbyM.Perepelitsa.Thecontributionof
Y.SunandZ.Zhangontheconditionalregularityofweaksolutionsprovidesalink
betweenthetheoryofweaksolutionsontheonehandandstrongsolutions/blowup
criteriaontheotherhand.
Three chapters are devoted to the stationary solutions, written by S. Jiang and
C.Zhou,byP.Mucha,M.Pokorný,andE. Zatorska,andbyP.Mucha,M.Pokorný,
andO.Kreml.
Compressible Navier-Stokes equations are applicable to modeling of a large
variety of fluid motions ranging from small-scale motions (as acoustic waves) to
large-scale motions of planetary size. The specific regimes of some of these flows
aredescribedsufficientlybysimplifiedmodelscharacterizedbyextremevaluesof
nondimensionalnumbers(asMach,Reynolds,Péclet,Strouhal,andothernumbers).
Some of the simplified models can be obtained rigorously from the compressible
Navier-Stokes equations as singular limits of nondimensional numbers. Typically,
theincompressibleNavier-StokessystemisknowntobealowMachnumberlimitof
thecompressibleNavier-Stokesequations.ThreehandbookchaptersbyN.Jiangand
N. Masmoudi,byFeireisl,andbyR.Kleinaredevotedtotheinvestigationofthe