ABOUT TWO TYPES OF MICROSTRUCTURES ADAPTED TO HEAT EVACUATION AND ELASTIC STRESS: SNOW FLAKES AND QUASI-CRYSTALS

doi:10.1016/S0252-9602(12)60006-X

摘要/Abstract

摘要：

I first met Constantine Dafermos in August 1974, at a meeting at Brown University, where I was invited because my former advisor (Jacques-Louis LIONS) could not come, and he had proposed my name. I was happily surprised that Constantine greeted me as if he knew me well, and since for many years now I have considered him as if he was an older brother, I wonder when this feeling started.
I do not remember when I told him that my paternal grandmother (Cecilia) had a Greek passport, which was not uncommon for Christian families in Syria, since most of them followed the Greek Orthodoxrite, and I now wonder if it played a role in the divorce of my grandparents, after my grandfather (Elias) switched to a more evangelical faith, and became a little excessive about religion: my father (Georges) once mentioned that when he had some argument with his father and he wanted to annoy him, he went to the Sunday offce at the Greek Orthodox¹ church .
Becoming friend with Constantine was natural for human reasons, but also for scientific

Abstract:

Luc Tartar. ABOUT TWO TYPES OF MICROSTRUCTURES ADAPTED TO HEAT EVACUATION AND ELASTIC STRESS: SNOW FLAKES AND QUASI-CRYSTALS[J]. 数学物理学报(英文版), 2012, 32(1): 84-108.

Luc Tartar. ABOUT TWO TYPES OF MICROSTRUCTURES ADAPTED TO HEAT EVACUATION AND ELASTIC STRESS: SNOW FLAKES AND QUASI-CRYSTALS[J]. Acta mathematica scientia,Series B, 2012, 32(1): 84-108.

参考文献

[1] Amirat Y, Hamdache K, Ziani A. Homog′en′eisation d’′equations hyperboliques du premier ordre – Appli-cation aux milieux poreux. Ann Inst H Poincar′e Anal Non Lin′eaire, 1989, 6(5): 397–417

[2] Amirat Y, Hamdache K, Ziani A. Etude d’une ′equation de transport `a m′emoire. C R Acad Sci Paris S′erI Math, 1990, 311(11): 685–688

[3] Courant R, Friedrichs K O. Supersonic Flow and Shock Waves. New York: Interscience Publishers, Inc,
1948

[4] Dafermos C M. Hyperbolic conservation laws in continuum physics (Grundlehren der Mathematischen
Wissenschaften [Fundamental Principles of Mathematical Sciences], 325. Berlin: Springer-Verlag, 2000;
Second edition, Berlin: Springer-Verlag, 2005

[5] DiPernaRJ.The structureofsolutionstohyperbolic conservation laws//Nonlinearanalysisand mechanics:
Heriot-Watt Symposium, Vol IV. Res Notes in Math, 39. Boston, Mass London: Pitman, 1979: 1–16

[6] DiPerna R J. Convergence of approximate solutions to conservation laws. Arch Rational Mech Anal, 1983,
82(1): 27–70

[7] Feynman R. Surely you’re joking, Mr. Feynman!. Vintage UK, 1992

[8] Hopf E. On the right weak solution of the Cauchy problem for a quasilinear equation of first order. J Math
Mech, 1969/1970, 19: 483–487

[9] Lax P D. Hyperbolic systems of conservation laws II. Comm Pure Appl Math, 1957, 10: 537–566

[10] Lax P D. Invariant functionals of nonlinear equations of evolution//Proc Internat Conf on Functional Analysis and Related Topics (Tokyo, 1969). Tokyo: Univ of Tokyo Press, 1970: 240–251

[11] Lax P D. Shock waves and entropy//Contributions to Nonlinear Functional Analysis (Proc Sympos, Math
Res Center, Univ Wisconsin, Madison, Wis, 1971). New York: Academic Press, 1971: 603–634

[12] NishidaT.Nonlinearhyperbolic equations andrelated topicsinfluiddynamics. Publications Math′ematiques
d’Orsay, No 78-02. D′epartement de Math′ematique, Universit′e de Paris-Sud, Orsay, 1978

[13] Serre D. Syst`emes de lois de conservation. I. Hyperbolicit′e, entropies, ondes de choc. Fondations. [Foun-
dations] Paris: Diderot Editeur, 1996; Systems of conservation laws. 1. Hyperbolicity, entropies, shock

waves (Translated from the 1996 French original by I. N. Sneddon). Cambridge: Cambridge University Press, 1999

[14] Serre D. Syst`emes de lois de conservation. II. [Systems of conservation laws. II] Structures g′eom′etriques, oscillation et probl`emes mixtes. Fondations. [Foundations] Diderot Paris: Editeur, 1996; Systems of
conservation laws. 2. Geometric structures, oscillations, and initial-boundary value problems (Translated from the 1996 French original by I. N. Sneddon). Cambridge: Cambridge University Press, 2000

[15] SmollerJ.Shock Waves andReaction-Di?usion Equations. Grundlehren derMathematischen Wissenschaften

[Fundamental Principles of Mathematical Science], Vol 258. New York: Springer-Verlag, 1983

[16] Tartar L. Nonlinear constitutive relations and homogenization//Contemporary Developments in Contin-
uum Mechanics and Partial Di?erential Equations (Proc Internat Sympos, Inst Mat, Univ Fed Rio de Janeiro, Rio de Janeiro, 1977). North-Holland Math Stud, 30. Amsterdam, New York: North-Holland, 1978: 472–484

[17] Tartar L. Equations hyperboliques non lin′eaires. S′eminaire Goulaouic–Schwartz (1977/1978), Exp No 18. Ecole′Polytech, Palaiseau, 1978

[18] Tartar L. Compensated compactness and applications to partial di?erential equations//Nonlinear Analysis
and Mechanics: Heriot-Watt Symposium, Vol IV. Res Notes in Math, 39. Boston, Mass London: Pitman, 1979: 136–212

[19] Tartar L. Remarks on Homogenization. Homogenization and Effective Moduli of Materials and Media.
The IMA Volumes in Mathematics and its Applications, Vol 1. New York: Springer, 1986: 228–246

[20] Tartar L. Une introduction `a la th′eorie math′ematique des syst`emes hyperboliques de lois de conservation. Publicazioni 682, Istituto di Analisi Numerica, Pavia, 1989

[21] Tartar L. Nonlocal e?ects induced by homogenization//Partial Di?erential Equations and the Calculus of
Variations. Essays in Honor of Ennio De Giorgi II. Boston: Birkha¨user, 1989: 925–938

[22] Tartar L. Memory effects and homogenization. Arch Rat Mech Anal, 1990, 111(2): 121–133

[23] Tartar L. H-measures, a new approach for studying homogenisation, oscillations and concentration effects
in partial di?erential equations. Proc Roy Soc Edinburgh Sect A, 1990, 115(3/4): 193–230

[24] Tartar L. H-measures and small amplitude homogenization//Random Media and Composites (Leesburg,
VA, 1988). Philadelphia, PA: SIAM, 1989: 89–99

[25] Tartar L. Beyond Young measures. Meccanica, 1995, 30: 505–526

[26] Tartar L. Remarks on the homogenization method in optimal design problems//Homogenization and
Applications to Material Sciences. Proc Nice 1995, Gakuto Int Series, Math Sciences and Applications Vol
9. Tokyo, Japan: Gakkokotosho, 1997: 393–412

[27] Tartar L. An Introduction to the homogenization method in optimal design. Optimal Shape Design, Tr′oia,
Portugal, 1998: 47–156 Lecture Notes in Math, Vol 1740. Fondazione CIME. Berlin: Springer-Verlag; Centro Internazionale Matematico Estivo, Florence, 2000

[28] Tartar L. Compensation e?ects in partial di?erential equations. Rend Accad Naz Sci XL Mem Mat Appl
(5), 2005, 29(1): 395–453

[29] Tartar L. From Hyperbolic Systems to Kinetic Theory, A Personalized Quest. Lecture Notes of the Unione
Matematica Italiana, 6. Berlin Heidelberg: Springer/Bologna: UMI, 2008

[30] Tartar L. The General Theory of Homogenization: A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana, 7. Heidelberg: Springer Berlin/Bologna: UMI, 2010

[31] Truesdell C A. Rational Thermodynamics. New York: McGraw-Hill, 1969