REMARKS ON THE CONTRIBUTIONS OF CONSTANTINE M. DAFERMOS TO THE SUBJECT OF ONSERVATION LAWS

doi:10.1016/S0252-9602(12)60002-2

摘要/Abstract

摘要：

Constantine M. Dafermos has done extensive researchat the interface ofpartial di?erential equations and continuum physics. He is a world leader in nonlinear hyperbolic conservation laws, where he introduced several fundamental methods in the subject including the methods of relative entropy, generalized characteristics, and wave-fronttracking, as well as the entropy rate criterion for the selection of admissible wave fans. He has also made fundamental contributions on the mathematical theory of the equations of thermomechanics as it pertains in modeling and analysis of materials with memory, thermoelasticity, and thermoviscoelasticity. His work is distinctly characterized by an understanding of the fundamental issues of continuum physics and their role in developing new techniques of mathematical analysis.

Abstract:

Gui-Qiang G. Chen, Athanasios E. Tzavaras. REMARKS ON THE CONTRIBUTIONS OF CONSTANTINE M. DAFERMOS TO THE SUBJECT OF ONSERVATION LAWS[J]. 数学物理学报(英文版), 2012, 32(1): 3-14.

Gui-Qiang G. Chen, Athanasios E. Tzavaras. REMARKS ON THE CONTRIBUTIONS OF CONSTANTINE M. DAFERMOS TO THE SUBJECT OF ONSERVATION LAWS[J]. Acta mathematica scientia,Series B, 2012, 32(1): 3-14.

参考文献

[1] Berthelin F, Vasseur A. From kinetic equations to multidimensional isentropic gas dynamics before shocks.
SIAM J Math Anal, 2005, 36: 1807–1835

[2] Bianchini S. On the Riemann problem for non-conservative hyperbolic systems. Arch Rational Mech Anal,
2003, 166: 1–26

[3] Brenier Y, De Lellis C, Sz`ekelyhidi Jr L. Weak-strong uniqueness for measure-valued solutions. Comm
Math Physics, 2011, 305: 351–361

[4] Bressan A. Global solutions of systems of conservation laws by wave-front tracking. J Math Anal Appl,
1992, 170: 414–432

[5] Bressan A. Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem. Oxford:
Oxford University Press, 2000

[6] Chen G -Q, Frid H, Li Y. Uniqueness and stability of Riemann solutions with large oscillation in gas
dynamics. Commun Math Phys, 2002, 228: 201–217

[7] Chen G -Q, Chen J. Stability of rarefaction waves and vacuum states for the multidimensional Euler
equations. J Hyperbolic Di? Equ, 2007, 4: 105–122

[8] Christoforou C, Spinolo L. A uniqueness criterion for viscous limits of boundary Riemann problems. J
Hyperbolic Di? Equations, 2011, 8(3): 507–544

[9] Dafermos C M. Polygonal approximations of solutions of the initial value problem for a conservation law.
J Math Anal Appl, 1972, 38: 33–41

[10] Dafermos C M. The entropy rate admissibility criterion for solutions of hyperbolic conservation laws. J
Di? Equ, 1973, 14: 202–212

[11] Dafermos C M. Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by
the viscosity method. Arch Rational Mech Anal, 1973, 52: 1–9

[12] Dafermos C M. Generalized characteristics and the structure of solutions of hyperbolic conservation laws.
Indiana Univ Math J, 1977, 26: 1097–1119

[13] Dafermos C M. The second law of thermodynamics and stability. Arch Rational Mech Anal, 1979, 70:
167–179

[14] Dafermos C M. Quasilinear hyperbolic systems with involutions. Arch Rational Mech Anal, 1986, 94:
373–389

[15] Dafermos C M. Generalized characteristics in hyperbolic systems of conservation laws. Arch Rational
Mech Anal, 1989, 107: 127–155

[16] Dafermos C M. Admissible wave fans in nonlinear hyperbolic systems. Arch Rational Mech Anal, 1989,
106: 243–260

[17] Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics. 3rd ed. Grundlehren der Mathe-
matischen Wissenschaften, 325. Berlin: Springer–Verlag, 2010

[18] Dafermos C M. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete
Contin Dyn Systems, 2009, 23: 185–195

[19] DafermosCM,DiPernaR J.The Riemann problemforcertain classesof hyperbolic systemsofconservation
laws. J Di? Equ, 1976, 20: 90–114

[20] Dafermos C M, Geng X. Generalized characteristics in hyperbolic systems of conservation laws with special
coupling. Proc Royal Soc Edinburgh, 1990, 116A: 245–278

[21] DiPerna R J. Global existence of solutions to nonlinear hyperbolic systems of conservation laws. J Di?
Equ, 1976, 20: 187–212

[22] DiPerna R J. Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ Mah J, 1979, 28:
137–187

[23] Filippov A F. Di?erential Equations with Discontinuuous Righthand Sides. Dordrecht: Kluwer, 1988

[24] Glimm J. Solutions in the large for nonlinear hyperbolic systems of conservation laws. Comm Pure Appl
Math, 1965, 18: 697–715

[25] Hattori H. The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion.
Isothermal case. Arch Rational Mech Anal, 1986, 92: 247–263

[26] Hattori H. The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion.
Nonisothermal case. J Di? Equ, 1986, 65: 158–174

[27] Holden H, Risebro N H. Front Tracking for Hyperbolic Conservation Laws. New York: Springer-Verlag,
2002

[28] Hsiao L. The entropy rate admissibility criterion in gas dynamics. J Di? Equ, 1980, 38: 226–238

[20] Joseph K T, LeFloch P G. Boundary layers in weak solutions of hyperbolic conservation laws II. Commun
Pure Appl Anal, 2002, 1: 51–76

[30] Keyfitz B L, Kranzer H C. A viscosity approximation to a system of conservation laws with no classical
Riemann solution//Lecture Notes in Math, 1402. Berlin: Springer, 1989: 185–197

[31] Lax P D. Hyperbolic Systems of Conservation Laws II. Comm Pure Appl Math, 1957, 10: 537–566

[32] Lax P D.Shock waves and entropy//Zarantonello EH,ed. Contributions to NonlinearFunctional Analysis.
New York: Academic Press, 1971: 603–634

[33] Liu T P. The Riemann problem for general systems of conservation laws. J Di? Equ, 1975, 18: 218–234

[34] Liu T P. Admissible solutions of hyperbolic conservation laws. Mem Amer Math Soc, 1981, 30(240): 1–78

[35] Risebro N H. A front-tracking alternative to the random choice method. Proc Amer Math Soc, 1993, 117:
1125–1139

[36] Slemrod M. A limiting “viscosity” approach to the Riemann problem for materials exhibiting change of
phase. Arch Rational Mech Anal, 1989, 105: 327–365

[37] Trivisa K. A priori estimates in hyperbolic systems of conservation laws via generalized characteristics.
Comm Partial Ditfer Equ, 1997, 22: 235–267

[38] Truesdell C, Noll W. The Non-linear Field Theories of Mechanics. 3rd ed. Handbuch der Physik. Berlin:
Springer-Verlag, 1965

[39] Tzavaras A E. Wave interactions and variation estimates for self-similar zero-viscosity limits in systems of
conservation laws. Arch Rational Mech Anal, 1996, 135: 1–60

[40] Tzavaras A E. Relative entropy in hyperbolic relaxation. Commun Math Sci, 2005, 3: 119–132