Lecture Notes of 6 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. Luc Tartar From Hyperbolic Systems to Kinetic Theory A Personalized Quest ABC LucTartar DepartmentofMathematicalSciences CarnegieMellonUniversity Pittsburgh,PA15213-3890 USA [email protected] ISBN978-3-540-77561-4 e-ISBN978-3-540-77562-1 DOI10.1007/978-3-540-77562-1 LectureNotesoftheUnioneMatematicaItalianaISSNprintedition:1862-9113 ISSNelectronicedition:1862-9121 LibraryofCongressControlNumber:2007942545 MathematicsSubjectClassification(2000):35K05,35L45,35L60,35L65,35L67,35Q30,70F45,76A02, 76N15,76P05,82C22,82C40 (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 Dedicated to Robert DAUTRAY Hehelpedmeatacriticaltime,whenIcouldnolongerbeartherejectioninthe academicworld(partlyforhavingrefusedthecurrentmethodsoffalsifications, and partly because I was too interested in science for a mathematician), and he also guidedme inmy readingswhile I workedatCommissariata` l’E´nergie Atomique, so that I did not get lost like many other mathematicians in the jungleofmodelswhichphysicistshavegenerated,andIcouldunderstandwhat mathematical tools should be developed for helping understand in a better way how nature works. to Peter LAX He gaveanexampleofhowagoodmathematiciancanwork,byputting some order in a corner of the physical world where the preceding knowledge was made up of a few examples and too many guesses. Why have there been so few mathematicians who wanted to follow his example? to Lucia to my children, Laure, Micha¨el, Andr´e, Marta and to my grandson, Lilian Preface After publishing An Introduction to Navier–Stokes Equation and Oceanogra- phy [20],1,2 and An Introduction to Sobolev Spaces and Interpolation Spaces [21],3 the revisedversions of my lecture notes for graduatecourses that I had taught in the spring of 1999 and in the spring of 2000, I want to follow with anothersetoflecture notesfora graduatecoursethatI hadtaughtinthe fall of 2001, with the title “Introduction to kinetic theory”. For this one, there hadbeen no versionavailableon the Internet, andI hadnotevenwritten the notes for the last four lectures, and after a few years,I find it useful to make the text available to a larger audience by publishing a revised and completed version, but I had to change the title in a significant way. In [21], I had written that my reasons for publishing lecture notes is to tell the readers some of what I have understood, the technical mathematical aspects of the course, the scientific questions behind the theories, and more, andIshallhavesucceededifmanybecomeaware,andgoforwardonthepath ofdiscovery,notmistakingresearchanddevelopment,knowingwhenandwhy theydooneortheother,andkeepingahighergoalinmindwhenforpractical reasonstheydecidetoobeythemottooftheageforawhile,publish or perish. In the fall of 2001,I had done precisely that, and I had taught the math- ematical results that I had proven during my quest for understanding about 1 Claude Louis Marie Henri NAVIER, French mathematician, 1785–1836. He had worked in Paris, France. He introduced the equation now known as the Navier– Stokes equation in 1821, although hedid not understand about shear stress. 2 SirGeorgeGabrielSTOKES,Irish-bornmathematician,1819–1903.Hehadworked inLondon,and inCambridge, England, holdingtheLucasian chair(1849–1903). 3 SergeiL’vovichSOBOLEV,Russianmathematician,1908–1989. Hehadworkedin Leningrad,inMoscow, andin Novosibirsk,Russia.ThereisnowaSobolev Insti- tuteof Mathematics of theSiberian branchof theRussian Academyof Sciences, Novosibirsk,Russia.IfirstmetSergeiSOBOLEVwhenIwasastudent,inParisin 1969, and conversed with him in French, which he spoke perfectly (all educated Europeans at thebeginning of the20th centurylearned French). VIII Preface kinetic theory, which I had started in the early 1970s, but I had also taught aboutwhatiswrongwithkinetictheory,whichIhadstartedtounderstandin the early 1980s, and I had tried to teach a little about continuum mechanics and physics with the critical mind of a mathematician, so that the students couldunderstand whatwere the results ofmy detective work onthis particu- lar question of kinetic theory, and understand how to attack other questions of continuum mechanics or physics by themselves later (having in mind the defects that have already been found on each question, by me or by others). In[21],I hadsuggestedtothe readerswhoalreadyknowsomething about continuummechanicsorphysicstolookatmylecturenotes,toreadaboutthe defects which I know about in classical models, because other authors rarely mention these defects even though they have heard about them. This set of lecture notes, written with a concern towards kinetic theory, is of this type. I had suggested to the readers who do not yet know much about continuum mechanicsorphysics,tostartwithmoreclassicaldescriptionsabouttheprob- lems, for example by consulting the books which have been prepared under the direction of Robert DAUTRAY,4 and of Jacques-Louis LIONS,5 whom he had convinced to help him, [5]–[13]. Ihavementionedthatmypersonalpointofview,whichisthatoneshould not follow the path of the majority when reason clearly points to a different direction,probablyowesalottohavingbeenraisedasthesonofa(Calvinist) Protestant minister,6 but I had lost the faith when I was twelve or thirteen years old, and I may not have explained well why I later found myself forced to practicethe artofthe detective indeciding whathadtobe discardedfrom what I could reasonably trust until some new information became available. Becoming a mathematician hadbeen one of the reasons,because mathemati- ciansmustknowwhatisprovenandwhatisonlyconjectured,andwhenlater I became interested in understanding continuum mechanics and physics from a mathematical point of view, I found that the analysis that must be done in organizing the information, as well as the misinformation that “scientists” transmit about the real world, is quite similar to the analysis that must be done in organizing the information and misinformation that various religious 4 Ignace Robert DAUTRAY (KOUCHELEVITZ), Frenchphysicist, born in 1928. 5 Jacques-Louis LIONS, French mathematician, 1928–2001. He received the Japan Prize in 1991. He had worked in Nancy and in Paris, France; he held a chair (analyse math´ematique des syst`emes et de leur controˆle, 1973–1998) at Coll`ege de France, Paris. The laboratory dedicated to functional analysis and numerical analysis which he initiated, funded by CNRS (Centre National de la Recherche Scientifique)andUniversit´eParisVI(PierreetMarieCurie),isnownamedafter him, the Laboratoire Jacques-Louis Lions. I first had Jacques-Louis LIONS as a teacher at E´cole Polytechnique in Paris in 1966–1967, and I did research under his direction, untilmy thesis in 1971. 6 Jean CALVIN (CAUVIN), French-born theologian, 1509–1564. He had worked in Paris and in Strasbourg, France, in Basel and in Gen`eve(Geneva), Switzerland. Preface IX traditions transmit, and in both these approaches, one can observe the per- verse influence of political factors. TheparticulardifficultythatIhadencounteredmyselfaround1980wasre- latedtothe politicalperversionofthe Frenchacademicsystemitself, because I found myself facing an unimaginable situation of forgeries, organized by a “mathematician” and continued by a “physicist”, which turned into a night- marewhenIwasrepeatedlyconfrontedwiththeracistbehaviourofthosewho insisted that it was normalthat I should not havethe same rights as others.7 Fortunately, Robert DAUTRAY provided me with a new job outside this strange “academic” world,8 and I was extremely grateful to him for that, as it contrasted a lot with the rejection that I was feeling in the mathematical world, including the strange opposition of my mentors, Laurent SCHWARTZ and Jacques-Louis LIONS,9 who had chosen the side of the forgers against me, probably because they had some different, wrong information. However, I am even more grateful to Robert DAUTRAY for something that very few people could have provided me, as my understanding of physics could not have improved in the way it did without his help, which was mostly through telling me what to read, and it is natural that I should dedicate this set of lecture notes to him, although he may not agree entirely with my personal analysis on the subject of kinetic theory. My new job, or more precisely what I had understood about what I had todo,hadbeenbothsimpleandimpossible,tounderstandphysicsinabetter way,throughamathematicalapproach,ofcourse.IfeltthatRobertDAUTRAY understood that physics had reached a few dead ends, where physicists were hitting some walls which had been created before them, by other physicists who had invented the wrong games for understanding how nature works. It shouldnothavebeentoocritical,asitisnaturalthatguessingproducesafew answers that are not completely right, although they may not be completely wrong,andusingtheartoftheengineeronecanmakethingsworkeventhough one does not have the correct equations for describing the processes that one wantsto tame, but this approachinscience has its limitations. Inorderto go forward, one needs to apply a scientific approach, and practice the art of the detective to discover what has been done wrong, and then one needs to do it in a better way, ideally in the right way, if that is possible. I thought that Robert DAUTRAY was not only aware of that, but that he saw that some of 7 ThishappenedinoneofthecampusesofUniversityParisXI(ParisSud),Orsay, France, from 1979 to 1982. 8 I worked at CEA (Commissariat `a l’E´nergie Atomique) in Limeil, France, from 1982 to 1987. 9 Laurent SCHWARTZ, French mathematician, 1915–2002. He received the Fields Medalin1950.HehadworkedinNancy,inParis,France,atE´colePolytechnique, whichwasfirstinParis (whenIhadhimasateacherin1965–1966), andthenin Palaiseau, and at Universit´eParis 7 (Denis Diderot), Paris. X Preface this work of providing more order must be done by mathematicians, at least well-trained mathematicians. Thejobofadetectiveiscertainlymadequitedifficultifhe/sheisforbidden toaskquestionstoimportantwitnesses,orifhe/sherealizesthatthereisawall ofsilenceandthatthereisinformationthatcouldbe usefulforhis/hersearch which some powerful group does not want him/her to discover. That type of difficultyexistsinphysics,aswellasinothersciences,includingmathematics. At the beginning, some guessed rule had been successful in one situation, and although it was dangerous to apply a similar guess indiscriminately for all kinds of problems, it had been done, but what made this practice quite unfortunatewasthentocreateadogma,andtoteachittonewgenerationsof students.Becausenohintsweregiventhatsomeoftheserulescouldbeslightly wrong,orevencompletelymisleading,thesephysicistswerenotreallytrained asscientists,anditisnotsurprisingthatmanyofthemendedupworkinglike engineers,mistakingphysicsandtechnology,andnotcaringmuchforthefact thatsomeofthe currentlytaught“lawsofphysics”areobviouslywrong:they aresimplythe lawsthatphysicistshaveguessedintheir questaboutthe laws that nature follows, and it would have been surprising that their first guess had been right. Before 1982, I had mostly thought about questions concerning contin- uum mechanics, developing homogenization and the compensated compact- ness method, partly with Franc¸ois MURAT,10 but I had also understood a questionof the appearanceof nonlocaleffects by homogenizationof some hy- perbolic equations, and I thought that this was a more rational explanation than the strange games of spontaneous absorption and emission that physi- cists had invented, so that their probabilistic games were just one possible approachto describing the correcteffective equations, confirming what I had alreadydiscoveredbefore,thatprobabilitiesareintroducedbyphysicistswhen theyfaceasituationthattheydonotunderstand,sothatitshouldbepointed outhowcrucialitistointroduceprobabilitiesaslateaspossibleinthe analy- sisofaproblem,ideallynotatallifpossible,butcertainlyfurtherandfurther awayfromonegenerationto thenext.However,upto1982,Ididnotseehow to include quantum mechanics and statistical mechanics in my approach to the partial differential equations of continuum mechanics and physics. After 1982, the first step was relatively easy, and in reading what Robert DAUTRAY had told me I identified a few points which are certainly wrong in thelawsthatphysicistsuse;however,makingthemrightseemedtorequirethe development of new mathematical tools. The tool of H-measures [18], which I started describing at the end of 1986, was something that I had already guessed two years before, but its extension to semi-linear hyperbolic systems 10 Franc¸oisMURAT,Frenchmathematician,bornin1947.HeworksatCNRS(Cen- treNationaldelaRechercheScientifique)andUniversit´eParisVI(PierreetMarie Curie), Paris, France. Preface XI has eluded me since, and I see that extension as necessary to explain some of the strange rules about quantum mechanics, and then derive better rules than those of statistical mechanics. At the end of 1983, a year before the first hint about new mathematical tools,Ialready“knew”whatiswrongwithkinetictheory,whichisthesubject of this set of lecture notes, as a consequence of having “understood” what is wrong with quantum mechanics. As I am a mathematician, I use quotes because I want to emphasize that it was not yet mathematical knowledge, anditwasnotaboutapreciseconjectureeitherbecauseIcouldnotformulate one at the time, but I had acquired the certitude that some aspects of what the physicists say will not appear in the new mathematical framework that I was searching for. The main mistake of physicists had been to stick to 18th century ideas of classical mechanics, instead of observing that if the 19th century ideas about continuum mechanics are inadequate for explaining what is observed at a microscopic level, it is because one needs new mathematical tools for 20thcenturymechanics/physics(turbulence,plasticity,atomicphysics),which have no probability in them, of course, as the use of probabilities is the sign that one does not understand what is going on. It had been a mistake to concentratetoomucheffortonproblemsofpartialdifferentialequationswhich show finite-dimensional effects, for which 18th century mechanics is adapted, instead of observing that the more interesting problems of partial differential equations all show infinite-dimensional effects, which cannot be grasped with 18th/19thcenturyideas;actually,mysubjectofresearchsincetheearly1970s hadbeenpreciselyfocusedonstudyingtheeffectofmicrostructuresinpartial differential equations, a subject which I have decided to describe as beyond partial differential equations. The certitude that mathematics brings is that thereareabsolutelyno particlesatatomic level,there areonlywaves,sothat there cannot be any particles interacting in the way that had been assumed by MAXWELL,11 and by BOLTZMANN.12 Nevertheless, one should be careful not to disparage MAXWELL and BOLTZMANN for the fact that their pioneering work in kinetic theory has some defects, because they had shown a good physical intuition for the way to correct an important defect of continuum mechanics, which is that the constitutive relations used are wrong, because they result from the inexact postulate that the relations valid at equilibrium are true at all times. That there are no particles and that they are waves could have been un- derstoodearlier,asaconsequenceofanobservationofPOINCARE´ inhisstudy 11 James CLERK MAXWELL, Scottish physicist, 1831–1879. He had worked in Aberdeen, Scotland, in London and in Cambridge, England, holding the first Cavendish professorship of physics(1871–1879). 12 LudwigBOLTZMANN,Austrianphysicist,1844–1906. HehadworkedinGrazand Vienna,Austria, in Leipzig, Germany, and then again in Vienna.