Navier-Stokes Equations : An Introduction with Applications

395 Pages·2016·2.22 MB·English

by  Grzegorz Łukaszewicz

Advances in Mechanics and Mathematics 34
Grzegorz Łukaszewicz
Piotr Kalita
Navier–
Stokes
Equations
An Introduction with Applications
Advances in Mechanics and Mathematics
Volume 34
SeriesEditors:
DavidY.Gao,VirginiaPolytechnicInstituteandStateUniversity
TudorRatiu,ÉcolePolytechniqueFédérale
AdvisoryBoard:
IvarEkeland,UniversityofBritishColumbia
TimHealey,CornellUniversity
KumbakonamRajagopal,TexasA&MUniversity
DavidJ.Steigmann,UniversityofCalifornia,Berkeley
Moreinformationaboutthisseriesathttp://www.springer.com/series/5613

Grzegorz Łukaszewicz (cid:129) Piotr Kalita
Navier–Stokes Equations
An Introduction with Applications
123
GrzegorzŁukaszewicz PiotrKalita
FacultyofMathematics, FacultyofMathematicsandComputer
Informatics,andMechanics Science
UniversityofWarsaw JagiellonianUniversityinKrakow
Warszawa,Poland Krakow,Poland
ISSN1571-8689 ISSN1876-9896 (electronic)
AdvancesinMechanicsandMathematics
ISBN978-3-319-27758-5 ISBN978-3-319-27760-8 (eBook)
DOI10.1007/978-3-319-27760-8
LibraryofCongressControlNumber:2015960205
©SpringerInternationalPublishingSwitzerland2016
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ToRenata,Agata,andJacek,withlove
(Grzegorz)
TomybelovedwifeKasia
(Piotr)

Preface
Admittedly,asusefulamatterasthemotionoffluidandrelated
scienceshasalwaysbeenanobjectofthought.Yetuntilthisday
neitherourknowledgeofpuremathematicsnorourcommandof
themathematicalprinciplesofnaturehaveasuccessful
treatment.
–DanielBernoulli
Incompressible Navier–Stokes equations describe the dynamic motion (flow) of
incompressiblefluid,theunknownsbeingthevelocityandpressureasfunctionsof
location(space)andtimevariables.Tosolvethoseequationswouldmeantopredict
thebehaviorofthefluidunderknowledgeofitsinitialandboundarystates.These
equationsareoneofthemostimportantmodelsofmathematicalphysics.Although
they have been a subject of vivid research for more than 150 years, there are still
many open problems due to the nature of nonlinearity present in the equations.
The nonlinear convective term present in the equations leads to phenomena such
as eddy flows and turbulence. In particular the question of solution regularity for
three-dimensionalproblemwasappointedbyClayMathematicsInstituteasoneof
theMillenniumProblems,thatis,thekeyproblemsinmodernmathematics.Thisis,
on one hand, due to the fact that the problem remains challenging and fascinating
for mathematicians and, on the other hand, that the applications of the Navier–
Stokes equations range from aerodynamics (drag and lift forces), through design
of watercrafts and hydroelectric power plants, to the medical applications of the
modelsofflowofbloodinvessels.
ThisbookisaimedatabroadaudienceofpeopleinterestedintheNavier–Stokes
equations,fromstudentstoengineersandmathematiciansinvolvedintheresearch
onthesubjectoftheseequations.
Itoriginatedinpartfromaseriesoflecturesofthefirstauthorgivenoverthepast
15yearsattheFacultyofMathematics,InformaticsandMechanicsoftheUniversity
of Warsaw; at summer schools at UNICAMP, Campinas, Brasil; and at Université
JeanMonnet,Saint-Etienne,France.Thelectureswerebasedontheleadingbooks
on the then young theory of infinite dimensional dynamical systems, focused on
mathematicalphysics,inparticular,onTemam[220];ChepyzhovandVishik[61];
Doering and Gibbon [88]; Foias¸, Manley, Rosa, and Temam [99]; and Robinson
[197].
vii
viii Preface
ThelecturesattheMathematicsFacultyoftheUniversityofWarsawwerealso
attended by students and PhD students from the Faculty of Physics and Faculty
of Geophysics, and it became clear that a routine mathematical lecture had to be
extendedtoincludeadditionalaspectsofhydrodynamics.Somestudentsaskedfor
“more physics and motivation” and “more real applications”; others were mainly
interestedinthemathematicsoftheNavier–Stokesequations,andyetotherswould
like to see the Navier–Stokes equations in a more general context of evolution
equationsandtolearnthetheoryofinfinitedimensionaldynamicalsystemsonthe
research level. These several aspects of hydrodynamics well suited the tastes and
interestsofthelecturer,andalsothesecondauthorwaswelcomedtojointheproject
ofthebookatalaterstage.
Inconsequence,theaudienceofthebookisthreesome:
GroupI:Mathematicians,physicists,andengineerswhowanttolearnaboutthe
Navier–Stokesequationsandmathematicalmodelingoffluids
GroupII:UniversityteacherswhomayteachagraduateorPhDcourseonfluid
mechanicsbasingonthisbookorhigher-levelstudentswhostartresearchonthe
Navier–Stokesequations
Group III: Researchers interested in the exchange of current knowledge on
dynamicalsystemsapproachtotheNavier–Stokesequations
Although,inprinciple,allthesethreegroupscanfindinterestinallchaptersof
thebook,Chaps.2–7areprimarilytargetedatGroupI,Chaps.3,4,7,8,11,and12
aimedmainlyatGroupII,andChaps.7–16forGroupIII.
For a reader with reasonable background on calculus, functional analysis, and
theoryofweaksolutionsforPDEs,thewholebookshouldbeunderstandable.
Thebookwasplannedtobeamonographwhichcouldalsobeusedasatextbook
to teach a course on fluid mechanics or the Navier–Stokes equations. Typical
coursescouldbe“Navier–Stokesequations”,“partialdifferentialequations”,“fluid
mechanics”, “infinite dimensional dynamics systems.” To this end many chapters
of this book include exercises. Moreover, we did not restrain ourselves to include
a number of figures to liven the text and make it more intuitive and less formal.
Webelievethatthefigureswillbehelpful.Specialcarewasundertakentokeepthe
individualchaptersself-containedasfaraspossibletoallowthereadertoreadthe
booklinearly(inlinearportions).Thatdemandedseveralsmallrepetitionshereand
there.
To understand the first chapters of this book, just the basic knowledge on
calculus,thatcanbelearnedfromanycalculustextbook,shouldbeenough.
The book is planned to be self-contained, but, to understand its last chapters,
someknowledgefromatextbooklike“PartialDifferentialEquations”byL.C.Evans
(which contains all necessary knowledge on functional analysis and PDEs) would
behelpful.Eachchaptercontainsanintroductionthatexplainsinsimplewordsthe
nature of presented resultsand asection onbibliographical notes thatwillplace it
inthecontextofpastandcurrentresearch.
Preface ix
Severalpeoplegreatlycontributed,knowinglyornotknowingly,tothecreation
ofthebook.Ourthanksgotoourcolleaguesandcollaborators:GuyBayada,Mahdi
Boukrouche,ThomasCaraballo,JoséLanga,PepeReal,JamesRobinsonetc.
ThefirstauthorisgratefultoGuyBayadawhointroducedhimtotheproblems
of lubrication theory and flows in narrow films during his visits at INSA, Lyon,
and to Mahdi Boukrouche with whom he collaborated for several years on this
subject.ThomasCaraballo,JoséLanga,andPepeRealintroducedhimtothesubject
of pullback attractors during his visit at the University of Seville. Thanks for the
opportunitytogivethesummercoursesinCampinasandSaint-EtiennegotoMarco
Rojas-MedarandMahdiBoukrouche,respectively.ManythanksgoalsotoChunyou
Sun, Meihua Yang, and Yongqin Xie for their kind invitation of the first author to
give several lectures at the University of Lanzhou, then at Huazhong University
of Science and Technology in Wuhan and at The University in Changsha, China,
in June 2013. Inspirational discussions and exchange of ideas with these Chinese
friends,includingalsoQingfengMaandYuejuanWang,contributedtotheformof
thelastchaptersofthebook.
The research group at Jagiellonian University with their leader, Stanisław
Migórski, greatly motivated the authors as regards contact problems. The second
author owes a lot to his colleagues and teachers from Jagiellonian University; he
would like to express his thanks especially to Zdzisław Denkowski and Stanisław
Migórski.HewouldalsoliketothankRobertSchaeferwhofirstintroducedtohim
thetopicsoffluidmechanics.Heisalsogratefulforinspiringdiscussionsinthefield
ofcontactmechanicstohiscollaboratorsfromtheUniversityofPerpignan,Mircea
SofoneaandMikaëlBarboteu.
We would like to thank Wojciech Pociecha for his help with the preparation of
thefigures.
TheworkwasinpartsupportedbytheNationalScienceCenterofPolandunder
theMaestroAdvancedProjectno.DEC-2012/06/A/ST1/00262.
Finally,weexpressourgratitudetotheAMMASerieseditor,DavidY.Gao,and
to Marc Strauss and the editors of Springer Publishing House for their care and
encouragementduringthepreparationofthebook.
Warszawa,Poland GrzegorzŁukaszewicz
Kraków,Poland PiotrKalita
October2015