Transport equations and multi-D hyperbolic conservation laws PDF
Preview Transport equations and multi-D hyperbolic conservation laws
Lecture Notes of 5 the Unione Matematica Italiana EditorialBoard FrancoBrezzi(EditorinChief) PersiDiaconis DipartimentodiMatematica DepartmentofStatistics UniversitadiPavia StanfordUniversity ViaFerrataI Stanford,CA94305-4065,USA 27100Pavia,Italy e-mail:[email protected], e-mail:[email protected] [email protected] JohnM.Ball NicolaFusco MathematicalInstitute DipartimentodiMatematicaeApplicazioni 24-29StGiles’ UniversitàdiNapoli“FedericoII”,viaCintia OxfordOX13LB ComplessoUniversitariodiMonteS.Angelo UnitedKingdom 80126Napoli,Italy e-mail:[email protected] e-mail:[email protected] AlbertoBressan CarlosE.Kenig DepartmentofMathematics DepartmentofMathematics PennStateUniversity UniversityofChicago UniversityPark 1118E58thStreet,UniversityAvenue StateCollege ChicagoIL60637,USA PA16802,USA e-mail:[email protected] e-mail:[email protected] FulvioRicci FabrizioCatanese ScuolaNormaleSuperiorediPisa MathematischesInstitut PlazzadeiCavalieri7 Universitatstraße30 56126Pisa,Italy 95447Bayreuth,Germany e-mail:[email protected] e-mail:[email protected] GerardVanderGeer CarloCercignani Korteweg-deVriesInstituut DipartimentodiMatematica UniversiteitvanAmsterdam PolitecnicodiMilano PlantageMuidergracht24 PiazzaLeonardodaVinci32 1018TVAmsterdam,TheNetherlands 20133Milano,Italy e-mail:[email protected] e-mail:[email protected] CédricVillani CorradoDeConcini EcoleNormaleSupérieuredeLyon DipartimentodiMatematica 46,alléed’Italie UniversitàdiRoma“LaSapienza” 69364LyonCedex07 PiazzaleAldoMoro2 France 00133Roma,Italy e-mail:[email protected] e-mail:[email protected] TheEditorialPolicycanbefoundatthebackofthevolume. Luigi Ambrosio • Gianluca Crippa Camillo De Lellis • Felix Otto Michael Westdickenberg Transport Equations and Multi-D Hyperbolic Conservation Laws Editors FabioAncona Stefano Bianchini Rinaldo M. Colombo Camillo De Lellis Andrea Marson Annamaria Montanari ABC Authors LuigiAmbrosio FelixOtto [email protected] [email protected] GianlucaCrippa MichaelWestdickenberg [email protected] [email protected] CamilloDeLellis [email protected] Editors FabioAncona CamilloDeLellis [email protected] [email protected] StefanoBianchini AndreaMarson [email protected] [email protected] RinaldoM.Colombo AnnamariaMontanari [email protected] [email protected] ISBN978-3-540-76780-0 e-ISBN978-3-540-76781-7 DOI10.1007/978-3-540-76781-7 LectureNotesoftheUnioneMatematicaItalianaISSNprintedition:1862-9113 ISSNelectronicedition:1862-9121 LibraryofCongressControlNumber:2007939405 MathematicsSubjectClassification(2000):35L45,35L40,35L65,34A12,49Q20,28A75 (cid:2)c 2008Springer-VerlagBerlinHeidelberg Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Coverdesign:WMXDesignGmbH Printedonacid-freepaper 987654321 springer.com Preface This book collects the lecture notes of two coursesand one mini-courseheld in a winterschoolinBolognainJanuary2005.Theaimofthisschoolwastopopularize techniques of geometric measure theory among researchers and PhD students in hyperbolicdifferentialequations. Thoughinitially developedin the contextof the calculusof variations,manyof these techniqueshave provedto be quite powerful forthetreatmentofsomehyperbolicproblems.Obviously,thispointofviewcanbe reversed:Wehopethatthetopicsofthesenoteswillalsocapturetheinterestofsome members of the elliptic community,willing to explore the links to the hyperbolic world. Thecourseswereattendedbyabout70participants(includingpost-doctoraland senior scientists) from institutions in Italy, Europe, and North-America. This ini- tiative was part of a series of schools (organized by some of the people involved intheschoolheldinBologna)thattookplaceinBressanone(Bolzano)inJanuary 2004, and in SISSA (Trieste) in June 2006. Their scope was to present problems and techniques of some of the most promising and fascinating areas of research related to nonlinearhyperbolicproblemsthat have received new and fundamental contributionsintherecentyears.Inparticular,theschoolheldinBressanoneoffered two courses that provided an introduction to the theory of control problems for hyperbolic-likePDEs (delivered by Roberto Triggiani), and to the study of trans- port equations with irregular coefficients (delivered by Francois Bouchut), while theconferencehostedinTriestewasorganizedintwocourses(deliveredbyLaure Saint-RaymondandCedricVillani)andinaseriesofinvitedlecturesdevotedtothe mainrecentadvancementsinthestudyofBoltzmannequation.Someofthemate- rialcoveredbythecourseofTriggianicanbefoundin[17,18,20],whilethemain contributions of the conference on Boltzmann will be collected in a forthcoming specialissueofthejournalDCDS,oftitle“Boltzmannequationsandapplications”. Thethreecontributionsofthepresentvolumegravitateallaroundthetheoryof BV functions, which play a fundamentalrole in the subject of hyperbolicconser- vationlaws.However,so farinthehyperboliccommunitylittle attentionhasbeen paidto sometypicalproblemswhichconstituteanoldtopicingeometricmeasure v vi Preface theory: the structure and fine properties of BV functions in more than one space dimension. The lecture notes of Luigi Ambrosio and Gianluca Crippa stem from the re- markable achievement of the first author, who recently succeeded in extending theso-calledDiPerna–LionstheoryfortransportequationstotheBVsetting.More precisely,considertheCauchyproblemforatransportequationwithvariablecoef- ficients ⎧ ⎨∂u(t,x)+b(t,x)·∇u(t,x) = 0, t (1) ⎩ u(0,x) = u (x). 0 WhenbisLipschitz,(1)canbeexplicitlysolvedviathemethodofcharacteristics: asolutionuisindeedconstantalongthetrajectoriesoftheODE ⎧ ⎨dΦx = b(t,Φ (t)) dt x (2) ⎩ Φ(0,x) = x. Transport equations appear in a wealth of problems in mathematical physics, whereusuallythecoefficientiscoupledtotheunknownsthroughsomenonlineari- ties.Thisalreadymotivatesfromapurelymathematicalpointofviewthedesireto developatheoryfor(1)and(2)whichallowsforcoefficientsbinsuitablefunction spaces.However,inmanycases,theappearanceofsingularitiesisawell-established centralfact:thedevelopmentofsuchatheoryishighlymotivatedfromtheapplica- tionsthemselves. Inthe1980s,DiPernaandLionsdevelopedatheoryfor(1)and(2)whenb∈W1,p (see[16]).ThetaskofextendingthistheorytoBVcoefficientswasalong-standing openquestion,untilLuigiAmbrosiosolveditin[2]withhisRenormalizationThe- orem. Sobolev functions in W1,p cannot jump along a hypersurface: this type of singularityisinsteadtypicalforaBVfunction.Therefore,notsurprisingly,Ambro- sio’s theoremhas foundimmediate application to some problemsin the theoryof hyperbolicsystemsofconservationlaws(see[3,5]). Ambrosio’sresult,togetherwithsomequestionsrecentlyraisedbyAlbertoBres- san,hasopenedthewaytoaseriesofstudiesontransportequationsandtheirlinks withsystemsofconservationlaws(see[4,6–13]).ThenotesofAmbrosioandCrippa containanefficientintroductionto the DiPerna–Lionstheory,a completeproofof Ambrosio’stheoremand an overviewof the furtherdevelopmentsand open prob- lemsinthesubject. The firstproofofAmbrosio’sRenormalizationTheoremreliesona deepresult ofAlberti,perhapsthedeepestinthetheoryofBVfunctions(see[1]). ConsideraregularopensetΩ⊂R2 andamapu:R2→R2 whichisregularin R2\∂Ωbutjumpsalongtheinterface∂Ω.Thedistributionalderivativeofuisthen thesumoftheclassicalderivative(whichexistsinR2\∂Ω)andasingularmatrix- valued radon measure ν, supportedon ∂Ω. Let μbe the nonnegativemeasure on R2 definedbythepropertythatμ(A)isthelengthof∂Ω∩A.Moreover,denoteby n the exterior unit normal to ∂Ω and by u− and u+, respectively,the interior and Preface vii exteriortracesofuon∂Ω.AsastraightforwardapplicationofGauss’theorem,we thenconcludethatthemeasureνisgivenby[(u+−u−)⊗n]μ. ConsidernowthesingularportionofthederivativeofanyBVvector-valuedmap. By elementaryresultsin measuretheory,we canalwaysfactorizeitintoa matrix- valued function M times a nonnegative measure μ. Alberti’s Rank-One Theorem statesthatthevaluesofMarealwaysrank-onematrices.Thedepthofthistheorem canbeappreciatedifonetakesintoaccounthowcomplicatedthesingularmeasure μcanbe. Thoughthe most recentproofof Ambrosio’sRenormalizationTheoremavoids Alberti’sresult,theRank-OneTheoremisapowerfultooltogaininsightinsubtle furtherquestions(seeforinstance[6]).ThenotesofCamilloDeLellisisashortand self-containedintroductiontoAlberti’sresult,wherethereadercanfindacomplete proof. Asalreadymentionedabove,thespaceofBVfunctionsplaysacentralroleinthe theoryofhyperbolicconservationlaws.ConsiderforinstancetheCauchyproblem forascalarconservationlaw ⎧ ⎨∂u+div [f(u)] = 0, t x (3) ⎩ u(0,·) = u . 0 It is a classical result of Kruzhkov that for bounded initial data u there exists a 0 uniqueentropysolutionto(3).Furthermore,ifu isafunctionofboundedvariation, 0 thispropertyisretainedbytheentropysolution. Scalar conservation laws typically develop discontinuities. In particular jumps along hypersurfaces, the so-called shock waves, appear in finite time, even when starting with smooth initial data. These discontinuities travel at a speed which can be computed through the so-called Rankine–Hugoniot condition. Moreover, the admissibility conditions for distributional solutions (often called entropy con- ditions)areinessencedevisedtoruleoutcertain“non-physical”shocks.Whenthe entropysolutionhasBVregularity,thestructuretheoryforBVfunctionsallowsus toidentifyajumpset,wherealltheseassertionsfindasuitable(measure-theoretic) interpretation. Whathappensifinsteadtheinitialdataaremerelybounded?Clearly,if f isalin- (cid:7)(cid:7) earfunction,i.e. f vanishes,(3)isatransportequationwithconstantcoefficients: extremelyirregularinitialdataarethensimplypreserved.Whenwearefarfromthis (cid:7)(cid:7) situation,looselyspeakingwhentherangeof f is“generic”, f iscalledgenuinely nonlinear. In one space dimension an extensively studied case of genuine nonlin- earityis that ofconvexfluxes f. Itis then an oldresult ofOleinik that, underthis assumption, entropy solutions are BV functions for any bounded initial data. The assumptionofgenuinenonlinearityimpliesaregularizationeffectfortheequation. In more than one space dimension (or undermilder assumptionson f) the BV regularizationnolongerholdstrue.However,Lions,Perthame,andTadmoregave in [19] a kinetic formulation for scalar conservation laws and applied velocity averagingmethodstoshowregularizationinfractionalSobolevspaces.Thenotesof GianlucaCrippa,FelixOtto,andMichaelWestdickenbergstartwithanintroduction viii Preface toentropysolutions,genuinenonlinearity,andkineticformulations.Theythendis- cusstheregularizationeffectsintermsoflinearfunctionspacesfora“generalized Burgers”flux,givingoptimalresults. Fromastructuralpointofview,however,theseestimates(eventheoptimalones) arealwaystooweaktorecoverthenicepictureavailablefortheBVframework,i.e. a solution which essentially has jump discontinuities behaving like shock waves. Guidedbytheanalogywiththeregularitytheorydevelopedin[14]forcertainvaria- tionalproblems,DeLellis,Otto,andWestdickenbergin[15]showedthatthispicture isanoutcomeofanappropriate“regularitytheory”forconservationlaws.Morepre- cisely,thepropertyofbeinganentropysolutiontoascalarconservationlaw(witha genuinelynonlinearflux f)allowsafairlydetailedanalysisofthepossiblesingular- ities.Theinformationgainedbythisanalysisisanalogoustothefinepropertiesof agenericBVfunction,evenwhentheBVestimatesfail.ThenotesofCrippa,Otto, andWestdickenberggiveanoverviewoftheideasandtechniquesusedtoprovethis result. ManyinstitutionshavecontributedfundstosupportthewinterschoolofBologna. WehadasubstantialfinancialsupportfromtheresearchprojectGNAMPA(Gruppo Nazionaleperl’AnalisiMatematica,laProbabilita`eleloroApplicazioni)–“Multi- dimensionalproblemsandcontrolproblemsforhyperbolicsystems”;fromCIRAM (ResearchCenterofAppliedMathematics)andtheFundforInternationalPrograms ofUniversityofBologna;andfromSeminarioMatematicoandtheDepartmentof MathematicsofUniversityofBrescia.Wewerealsofundedbytheresearchproject INDAM (Istituto Nazionale di Alta Matematica “F. Severi”) – “Nonlinear waves andapplicationstocompressibleandincompressiblefluids”.Ourdeepestthanksto alltheseinstitutionswhichmakeitpossibletherealizationofthiseventandasacon- sequenceof the presentvolume.As a finalacknowledgement,we wish to warmly thankAccademiadelleScienzediBolognaandthe DepartmentofMathematicsof Bolognafortheirkindhospitalityandforallthehelpandsupporttheyhaveprovided throughouttheschool. Bologna,Trieste, FabioAncona Brescia,Zu¨rich, StefanoBianchini andPadova, RinaldoM.Colombo September2007 CamilloDeLellis AndreaMarson AnnamariaMontanari References 1. ALBERTI,G.Rank-onepropertiesforderivativesoffunctionswithboundedvariationsProc. Roy.Soc.EdinburghSect.A,123(1993),239–274. 2. AMBROSIO,L.TransportequationandCauchyproblemforBV vectorfields.Invent.Math., 158(2004),227–260. Preface ix 3. AMBROSIO,L.;BOUCHUT,F.;DELELLIS,C.Well-posednessforaclassofhyperbolicsys- temsofconservationlawsinseveralspacedimensions.Comm.PartialDifferentialEquations, 29(2004),1635–1651. 4. AMBROSIO,L.; CRIPPA, G.; MANIGLIA,S. Traces and fine properties of a BD class of vectorfieldsandapplications.Ann.Fac.Sci.ToulouseMath.(6)14(2005),no.4,527–561. 5. AMBROSIO,L.; DE LELLIS, C.Existenceofsolutionsforaclassofhyperbolicsystemsof conservationlawsinseveralspacedimensions.Int.Math.Res.Not.41(2003),2205–2220. 6. AMBROSIOL,; DE LELLIS, C.; MALY´, J. Onthechainruleforthedivergence ofvector fields: applications, partial results, open problems. To appear in Perspectives inNonlinear PartialDifferentialEquations:inhonorofHaimBrezisPreprintavailableathttp://cvgmt.sns. it/papers/ambdel05/. 7. AMBROSIOL.;LECUMBERRY,M.;MANIGLIA,S.S.Lipschitzregularityandapproximate differentiabilityoftheDiPerna–Lionsflow.Rend.Sem.Mat.Univ.Padova114(2005),29–50. 8. BRESSAN,A.AnillposedCauchyproblemforahyperbolicsystemintwospacedimensions. Rend.Sem.Mat.Univ.Padova110(2003),103–117. 9. BRESSAN, A. A lemmaand aconjecture onthecost of rearrangements. Rend. Sem. Mat. Univ.Padova110(2003),97–102. 10. BRESSAN,A.Someremarksonmultidimensionalsystemsofconservationlaws.AttiAccad. Naz.LinceiCl.Sci.Fis.Mat.Natur.Rend.Lincei(9)Mat.Appl.15(2004),225–233. 11. CRIPPA,G.;DELELLIS,C.Oscillatorysolutionstotransportequations.IndianaUniv.Math. J.55(2006),1–13. 12. CRIPPA,G.;DELELLIS,C.EstimatesandregularityresultsfortheDiPerna-Lionsflow.To appear inJ.ReineAngew. 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LIONSP.-L.;PERTHAMEB.;TADMORE.Akineticformulationofmultidimensionalscalar conservationlawsandrelatedquestions.J.AMS,7(1994)169–191. 20. TRIGGIANI,R. Global exact controllability on HΓ1(Ω)×L2(Ω) of semilinear wave equa- 0 tionswithNeumann L (0,T;L (Γ ))-boundary control. In:Control theory of partial differ- 2 2 1 entialequations,273–336,Lect.NotesPureAppl.Math.,242,Chapman&Hall/CRC,Boca Raton,FL,2005.
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