Concepts of hyperbolicity and relativistic continuum mechanics

Lecture Notes in Physics

Volumn 692, Issue , 2006, Pages 101-116

  Beig, Robert ^a  

^a UNIVERSITY OF VIENNA   (Austria)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33748949895     PISSN: 00758450     EISSN: None     Source Type: Book Series    
DOI: 10.1007/3-540-33484-X_5     Document Type: Review

References (45)

1. D. Alekseevsky, A. Kriegl, M. Losik, P.W. Michor: Choosing roots of polynomials smoothly. Israel J. Math. 105, 203-233 (1998)
2. A.M. Anile: Relativistic Fluids and Magneto-Fluids (Cambride University Press, Cambridge 1989)
3. M.F. Atiyah, R. Bott, L. Gårding: Lacunas for hyperbolic differential operators with constant coefficients I; II. Acta Math. 124, 109-189 (1970); 131, 145-206 (1973)
4. S. Barcelo, S. Liberati, S. Sonego, M. Visser: Causal structure of acoustic space-times (2004) gr-qc/04080221
5. H.H. Bauschke, O. Güler, A.S. Lewis, H.S. Sendov: Hyperbolic polynomials and convex analysis. Can. J. Math. 53, 470-488 (2001)
6. R. Beig, B.G. Schmidt: Relativistic elasticity. Class. Quantum Grav. 20, 889-904 (2003)
7. R. Beig: Nonrelativistic and relativistic continuum mechanics. To appear in the Proceedings of the Seventh Hungarian Relativity Workshop, Sarospatak, Hungary. gr-qc/0403073
8. R. Beig, M. Wernig-Pichler: The Cauchy problem for relativistic elastic bodies with natural boundary conditions. In preparation (2004)
9. R. Benedetti, J.-J. Risler: Real Algebraic and Semialgebraic Sets (Hermann, Paris 1990)
10. D. Christodoulou: The Action Principle and Partial Differential Equations (Princeton University Press, Princeton 2000)
11. D. Christodoulou: On hyperbolicity. Contemporary Mathematics 283, 17-28 (2000)
12. P.T. Chruściel: Black holes. In Conformal Structure of Spacetime, ed by J. Frauendiener and H. Friedrich, Lecture Notes in Physics 604 (Springer, Heidelberg 2002)
13. R. Courant, D. Hilbert: Methods of Mathematical Physics, Volume II (Interscience Publishers, New York 1962)
14. D.M. DeTurck, D. Yang: Existence of elastic deformations with prescribed principal strains and triply orthogonal systems. Duke Math. Journal 51, 243-260 (1984)
15. G.F.D. Duff: The Cauchy problem for elastic waves in an anisotropic medium. Transactions Roy. Soc. A 252, 249-273 (1960)
16. J. Frauendiener: A note on the relativistic Euler equations. Class. Quantum Grav. 20, L193-L196 (2003)
17. H. Friedrich: On the regular and the asymptotic characteristic initial value problem for Einstein's vacuum field equations. Proc. Roy. Soc. A 375, 169-184 (1985)
18. H. Friedrich, A. Rendall: The Cauchy problem for the Einstein equations. In: Einstein's Field Equations and their Physical Interpretation, ed by B. G. Schmidt, Lecture Notes in Physics 540 (Springer, Heidelberg 2000)
19. L. Gårding: An inequality for hyperbolic polynomials. J. of Mathematics and Mechanics 8, 957-965 (1959)
20. L. Gårding: History of the mathematics of double refraction. Arch. Hist. Ex. Sci. 40, 355-385 (1989)
21. I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky: Discriminants, Resultants and Multidemensional Determinants (Birkhäuser, Boston 1994)
22. M. Giaquinta, S. Hildebrandt: Calculus of Variations II (Springer, Berlin 1996)
23. G.-M. Greuel, G. Pfister, H. Schönemann: Singular 2.0. A computer algebra system for polynomial computations. Centre for Computer Algebra, University of Kaiserslautern (2001) http://www.singular.uni-kl.de
24. F.W. Hehl, Y.N. Obukhov: Foundations of Classical Electrodynamics - Charge, Flux, and Metric (Birkhäuser, Basel 2003)
25. G. Herglotz: Vorlesungen über die Mechanik der Kontinua. Göttingen lectures of 1926 and 1931, elaborated by R. B. Guenther and H. Schwerdtfeger (Teubner, Leipzig 1988)
26. L. Hörmander: Hyperbolic systems with double characteristics. Comm. Pure Appl. Math. 46, 89-106 (1993)
27. T.J.R. Hughes, T. Kato, J.E. Marsden: Well-posed quasilinear second-order hyperbolic systems with applications to Nonlinear Elastodynamics and General Relativity. Arch. Rat. Mech. Anal. 63, 273-294 (1977)
28. F. John: Algebraic conditions for hyperbolicity of systems of partial differential equations. Comm.Pure Appl.Math. 31, 89-106 (1978)
29. T. Kato: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Rat. Mech. Anal. 58, 181-205 (1975)
30. A. Kostelecký, M. Mewes: Signals for Lorentz violation in electrodynamics. Phys. Rev. D 66, 056005-056028 (2002)
31. H.-O. Kreiss, O. Ortiz: Some mathematical and numerical questions connected with first and second order time dependent systems of partial differential equations. In: The Conformal Structure of Space-Time, ed by J. Frauendiener and H. Friedrich), Lecture Notes in Physics 604 (2002)
32. C. Lämmerzahl, F. W. Hehl: Riemannian cone from vanishing birefringence in premetric electrodynamics. (2004) gr-qc/0409072
33. G. Nagy, O.E. Ortiz, O.A. Reula: Strongly hyperbolic second order Einstein's evolution equations. Phys. Rev. D 70, 044012(15) (2004)
34. V. Perlick: Ray Optics, Fermat's Principle, and Applications to General Relativity, Lecture Notes in Physics m61 (Springer, Heidelberg 2000)
35. J. Rauch: Group velocity at smooth points of hyperbolic characteristic varieties. Astérisque 284, 265-269 (2003)
36. G.F. Rubilar: Linear pre-metric electrodynamics and deduction of the light-cone. Ann. Phys. (Leipzig) 11, 717-782 (2002)
37. G. Salmon: Lectures Introductory to the Modern Higher Algebra (Dublin 1885; reprinted by Chelsea Publ., New York 1969)
38. D. Serre: Formes quadratiques et calcul des variations. J. Math. Pures et Appl. 62, 177-196 (1983)
39. M. Stoth: Decay estimates for solutions of linear elasticity for anisotropic media. Math. Meth. Appl. Sci. 19, 15-31 (1996)
40. M.E. Taylor: Pseudodifferential Operators and Nonlinear PDE (Birkhäuser, Boston 1991)
41. M.E. Taylor: Pseudodifferential Operators (Princeton University Press, Princeton 1981)
42. F.J. Terpstra: Die Darstellung biquadratischer Formen als Summen von Quadraten mit Anwendung auf die Variationsrechnung. Math. Ann. 116, 166-180 (1938)
43. A. Trautman: Comparison of Newtonian and relativistic theories of space-time. In: Perspectives in Geometry and Relativity, ed by B. Hoffmann, (Indiana University Press 1966)
44. J.P. Wolfe: Imaging Phonons (Cambridge University Press, Cambridge 1998)
45. A.Ç. Zenginoglǔ: Ideal magnetohydrodynamics in curved spacetime. Masters Thesis, University of Vienna (2003)