From hyperbolic systems to kinetic theory: a personalized quest

294 Pages·2008·2.126 MB·English

by  Luc Tartar

Lecture Notes of 6
the Unione Matematica Italiana
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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
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MathematicsSubjectClassification(2000):35K05,35L45,35L60,35L65,35L67,35Q30,70F45,76A02,
76N15,76P05,82C22,82C40
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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.