Lecture Notes in Mathematics 2254 CIME Foundation Subseries Matthias Hieber James C. Robinson  Yoshihiro Shibata Mathematical Analysis of the Navier-Stokes Equations Cetraro, Italy 2017 Giovanni P. Galdi · Yoshihiro Shibata Editors Lecture Notes in Mathematics 2254 Editors-in-Chief: Jean-MichelMorel,Cachan BernardTeissier,Paris AdvisoryEditors: KarinBaur,Leeds MichelBrion,Grenoble CamilloDeLellis,Princeton AlessioFigalli,Zurich AnnetteHuber,Freiburg DavarKhoshnevisan,SaltLakeCity IoannisKontoyiannis,Cambridge AngelaKunoth,Cologne ArianeMézard,Paris MarkPodolskij,Aarhus SylviaSerfaty,NewYork GabrieleVezzosi,Florence AnnaWienhard,Heidelberg Moreinformationaboutthissubseriesathttp://www.springer.com/series/3114 Matthias Hieber (cid:129) James C. Robinson (cid:129) Yoshihiro Shibata Mathematical Analysis of the Navier-Stokes Equations Cetraro, Italy 2017 Giovanni P. Galdi (cid:129) Yoshihiro Shibata Editors Authors MatthiasHieber JamesC.Robinson Department(FB)ofMathematics MathematicsInstitute TechnischeUniversita¨tDarmstadt UniversityofWarwick Darmstadt,Germany Coventry,UK YoshihiroShibata DepartmentofMathematics WasedaUniversity Tokyo,Japan Editors GiovanniP.Galdi YoshihiroShibata MEMSDepartment DepartmentofMathematics UniversityofPittsburgh WasedaUniversity Pittsburgh,PA,USA Tokyo,Japan ISSN0075-8434 ISSN1617-9692 (electronic) LectureNotesinMathematics C.I.M.E.FoundationSubseries ISBN978-3-030-36225-6 ISBN978-3-030-36226-3 (eBook) https://doi.org/10.1007/978-3-030-36226-3 MathematicsSubjectClassification(2010):Primary:35Q30,76D05,35Q35;Secondary:65Mxx,65Nxx ©SpringerNatureSwitzerlandAG2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface As is well known, the Navier–Stokes equations constitute one of the most active and attractive areas of research, from both theoretical and applied viewpoints. In particular, especially over the last two decades, they have been the focus of a number of fundamental mathematical contributions from different perspectives. Probably,thisrapidgrowthmaybealsoduetothecircumstancethat,sincetheyear 2000,thequestionofexistenceofglobal,regularsolutionscorrespondingtoinitial data of unrestricted size has been declared as one of the main open Millennium Problems by the Clay Mathematical Institute. Yet, in spite of all efforts, to date, theinvolvedmathematiciansunanimouslyagreethatverylittleisstillknownabout theseequationsandthattheirdeepsecretsarestillfarfrombeinguncovered. The principal objective of the CIME school on “Mathematical Analysis of the Navier–StokesEquations:FoundationsandOverviewofBasicOpenProblems,”in Cetraro,September4–8 2017,was to provideseries of lecturesdevotedto several fundamental and diverse aspects of the Navier–Stokes equations. The present volumecollectssomeofthem,includingboundarylayers,fluid–solidinteractions, freesurfaceandcomplexfluidproblems(ProfessorMatthiasHieber,TUDarmstadt, Germany); questions of existence, uniqueness, and regularity (Professor James C. Robinson, University of Warwick, UK); and local and global well-posedness andasymptoticbehaviorforfreeboundaryproblems(ProfessorYoshihiroShibata, WasedaUniversity,Japan). It is our distinct pleasure to thank all lecturers and participants for their enthusiastic—scientific and social—contribution to the success of the school, the Fondazione CIME and its scientific committee for giving us the opportunity to organizethisevent,andallthestafffortheirinvaluablehelp. Pittsburgh,PA,USA GiovanniP.Galdi Tokyo,Japan YoshihiroShibata v Contents 1 AnalysisofViscousFluidFlows:AnApproachbyEvolution Equations ..................................................................... 1 MatthiasHieber 2 PartialRegularityforthe3DNavier–StokesEquations................. 147 JamesC.Robinson 3 R Boundedness, MaximalRegularityand FreeBoundary ProblemsfortheNavierStokesEquations................................ 193 YoshihiroShibata vii Chapter 1 Analysis of Viscous Fluid Flows: An Approach by Evolution Equations MatthiasHieber Preface Thiscourseoflecturesdiscussesvariousaspectsofviscousfluidflowsrangingfrom boundarylayers andfluid structureinteractionproblemsoverfree boundaryvalue problems and liquid crystal flow to the primitive equations of geophysical flows. We will be mainly interested in strong solutions to the underlying equations and chooseasmathematicaltoolforourinvestigationsthetheoryofevolutionequations. The models considered are mainly represented by semi- or quasilinear parabolic equations and from a modern point of view it is hence natural to investigate the underlyingequationsbymeansofthemaximalLp-regularityapproach. For this reason, we start these lectures by an introduction to Cauchy problems andsectorialoperators.Thelatter are thestarting pointforthe functionalcalculus of bounded, holomorphicfunctions, which will be then extended to the operator- valued H∞-calculus. The extension leads us to the notion ofR-boundedfamilies ∞ of operators, which will be the key for boundedness results of the H -calculus in this setting. It implies the Kalton–Weis theorem on the closedness of the sum oftwo commuting,sectorialoperatorsandvia the extensionofMikhlin’stheorem to Banach spaces having the UMD-property, also the characterization theorem of maximalLp-regularityforparabolicevolutionequationsintermsofR-boundedness of its resolvent. We then proceed with quasilinear parabolic evolution equations and present three important results for these equations: local well-posedness, the generalized principle of linearized stability implying under suitable assumptions theglobalexistenceofstrongsolutionsfordataclosetoanequilibriumpointandon theexistenceofglobalstrongsolutionsinthepresenceofcompactembeddingsand strictLyapunovfunctionals. M.Hieber((cid:2)) Department(FB)ofMathematics,TUDarmstadt,Darmstadt,Germany e-mail:[email protected] ©SpringerNatureSwitzerlandAG2020 1 G.P.Galdi,Y.Shibata(eds.),MathematicalAnalysisoftheNavier-Stokes Equations,LectureNotesinMathematics2254, https://doi.org/10.1007/978-3-030-36226-3_1 2 M.Hieber Comingbacktothemainaimstothesenotes,well-posednessresultsforviscous fluid flows, we start the second partwith a discussion of balance laws for general heatconductingfluidsanddeducefromtherethefundamentalequationsofviscous fluid flows, the incompressible and compressible Navier–Stokes equations. We continue with the analysis of the Stokes equation in a half space Rn+ within the Lp-setting. The associated operator, the negative Stokes operator, is shown to be a sectorial operator on Lp(Rn+). As a consequence of the results in Part A we obtain the propertyof maximal Lp–Lq-regularity for the Stokes equation on Rn+. A localization procedure yields then the corresponding regularity results for the Stokesequationsonstandarddomains. We then consider stability questions for Ekman boundary layers. The latter are explicit stationary solutions of the Navier–Stokes equations in the rotational framework. We then show that the Ekman layer is asymptotic stable provided the Reynolds number involved is small enough. We continue with moving and free boundaryvalue problems. In fact, a fluid-rigid body interaction problem will be discussed for the situation of compressible fluids. Maximal regularity of the linearized equation in Lagragian coordinates allows us to prove a local well- posedness result for strong solutions. Strong solutions for the two-phase problem for generalized Newtonian fluids are again obtained by a fixed point argumentin the associated space of maximal regularity. We finally study also the primitive equations, which are a model for oceanic and atmospheric flows and are derived from the Navier–Stokes equations by assuming a hydrostatic balance for the pressure term. We show that these equationsare globally stronglywell-posedness for arbitrary large initial data lying in critical spaces. Finally, we show that the primitiveequationsmaybe obtainedasthelimitofanisotropicallyscaled Navier– Stokes equations. The approach to the results concerning the primitive equations are based again on maximal Lp-regularity estimates, this time for the hydrostatic Stokesoperator. My sincere thanksgo to the FondazioneCIME for their verykind supportand hospitality during a Summer Course on ‘Mathematical Analysis of the Navier– Stokes Equations: Foundations and Overview of Basic Open Problems’ held at CetraroinSeptember2017,inwhichtheselecturesnotesweredeveloped. PartA:ParabolicEvolutionEquations Thispartofthesenotesmainlyconcernsthetheoryofevolutionequations.Aiming for applications to viscous fluid flows we are mainly interested in parabolic evolutionequations. Westartbyintroducingthebasicconceptofdistributions,theSchwartzspaceS and its dual space S(cid:3), the Fourier transform on S and S(cid:3) and discuss also briefly varioustypesoffunctionspacesandtheirbasicproperties.Furthermore,wedefine Fourier multipliers for Lp(Rn). The celebrated theorem due to Mikhlin on Lp- 1 AnalysisofViscousFluidFlows:AnApproachbyEvolutionEquations 3 boundednessoftranslationsinvariantoperatorsonLp(Rn)for1 < p < ∞aswell as its analoguein the periodsetting will be used later on in manyoccasions.This theorem allows us further to introduce the Hilbert transform as well as the Riesz transformsasboundedoperatorsonLp(R)andLp(Rn),respectively. In Sect.1.2 we generalize this concept to the vector-valued setting, i.e. given a Banach space X, we discuss X-valued distributions, the Bochner integral and basictheoremsforsingularintegraloperatorswithoperator-valuedkernels.Banach spaceshavingthe UMD-propertywillbe defindedas spacesfor whichtheHilbert transformactsasaboundedoperatoronLp(R;X)forsomep ∈(1,∞). In the sequel, we consider in Sect.1.3 semigroups of operators and their generators. The classical theorems due to Hille-Yosida and Lumer-Philipps will be proven as well as the characterization theorem and smoothing properties for holomorphicsemigroups. In Sect.1.4we introducesectorialoperators.Theyarethe startingpointforthe functionalcalculusofbounded,holomorphicfunctions,which,includingimportant ∞ examples,willbeinvestigatedinthissection.AnextendendH -calculusallowsus todefinefractionalpowersofsectorialoperatorsandtheirpropertiesinanelegant way. ∞ Section 1.6 deals with the operator-valued H -calculus. The extension of the scalar-valuedH∞-calculusto theX-valuedfunctionsleadsusto the notionof R- bounded families of operators. This notion will be the key for the boundedness ∞ result of the H -calculus in this setting. It implies the Kalton–Weis theorem on the closedness of the sum of two commuting, sectorial operators and via the extension of Mikhlin’s theorem to Banach spaces having the UMD-property,also the characterization theorem of maximal Lp-regularity for parabolic evolution equationsintermsofR-boundednessofitsresolvent. The final section of this first part discusses quasilinear parabolic evolution equations. Here we present the approach based on the theory of maximal Lp- regularity.Thispropertywillbethekeyproperty,whenpresentingthreeimportant results for these equations: local well-posedness, the generalized principle of linearizedstabilityimplyingundersuitableassuptionstheglobalexistenceofstrong solutionsfordataclosetoanequilibriumpointandontheexistenceofglobalstrong solutionsinthepresenceofcompactembeddingsandstrictLyapunovfunctionals. 1.1 Basics onDistributions,Fourier Transforms andSobolev Spaces In this section we collect basic facts on distributions, the Fourier transforms and functionspacesandalsointroducethenotationbeingusedlateron. DistributionsandFourierTransforms Letusbeginwiththenotionofamultiindex.Amultiindexα =(α ,...,α )∈Nnis 1 n 0 anorderedn-tupleofnonnegativeintegersandwedenoteby|α|=α +...+α the 1 n