Transport equations and multi-D hyperbolic conservation laws

138 Pages·2008·0.926 MB·English

by  Luigi Ambrosio, Gianluca Crippa, Felix Otto, Michael Westdickenberg, Camillo De Lellis, Camillo De Lellis, Fabio Ancona, Stefano Bianchini, Rinaldo M. Colombo, Andrea Marson, Annamaria Montanari

Lecture Notes of 5
the Unione Matematica Italiana
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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
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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
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