Geometric singularities for solutions of single conservation laws

Archive for Rational Mechanics and Analysis

Volumn 139, Issue 3, 1997, Pages 255-290

  Izumiya, Shyuichi ^a   Kossioris, Georgios T ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031502309     PISSN: 00039527     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002050050053     Document Type: Article

References (24)

1. V. I. ARNOL'D, Geometric Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, 1983.
2. D. P. BALLOU, Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions, Trans. Amer. Math. Soc. 152 (1970), 441-460.
3. I. A. BOGAEVSKII, Modifications of singularities of minimum functions and bifurcations of shock waves af the Burgers equation with vanishing viscosity, Leningrad Math. J. 1 (1990), 807-823.
4. Y. BRENIER, Averaged multi-valued solutions for scalar conservation laws, SIAM J. Numer. Anal. 21 (1984), 1013-1037.
5. R. E. CAFLISCH, N. ERCOLANI, T. Y. HOU & Y. LANDIS, Multi-valued solutions and branch point singularities for nonlinear hyperbolic svstems, Comm. Pure Appl. Math. 46 (1993), 453-499.
6. N. M. CHEN, On types of singularities for solutions of nonlinear hyperbolic systems, Bull. Inst. Math. Acad. Sinica 10 (1982), 405-416.
7. C. M. DAFERMOS, Regularity and large time behaviour of solutions of a conservation law without convexity, Proc. Roy. Soc. Edinburgh 99A (1985), 201-239.
8. C. M. DAFERMOS, Generalized characteristics in hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 107 (1989), 127-155.
9. J. DAMON, The unfolding and determinacy theorems for subgroups of script A sign and script K sign, Memoirs of Amer. Math. Soc. 50-306 (1984).
10. J. GLIMM, D. MARCHESIN & O. MCBRYAN, Unstable fingers in two phase flows, Comm. Pure Appl. Math. 34 (1981), 53-75.
11. M. GOLUBITSKY & V. GUILLEMIN, Stable Mappings and Their Singularities, Springer-Verlag, 1973.
12. J. GUCKENHEIMER, Solving a single conservation law, Lecture Notes in Mathematics 468, Springer Verlag, New York, (1975), pp. 108-134.
13. S. IZUMIYA, Geometric singularities for Hamilton-Jacobi equation, Adv. Stud. Pure Math. 22 (1993), 89-100.
14. S. IZUMIYA & G. T. KOSSIORIS, Semi-local classification of geometric singularities for Hamilton-Jacobi equations, J. Diff. Eqs. 118 (1995), 166-193.
15. G. JENNINGS, Piecewise smooth solutions of a single conservation law exist, Adv. in Math, 33(1979), 192-205.
16. S. N. KRUŽKOV, First order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), 217-243.
17. P. LIONS, B. PERTHAME & E. TADMOR, A kinetic formulation of mutltidimension alscalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994) 169-191.
18. J. MARTINET, Singularities of Smooth Functions and Maps, Cambridge University Press, 1982.
19. S. NAKANE, Formation of shocks for a single conservation law, SIAM J. Math. Anal. 19 (1988), 1391-1408.
20. A. KH. RAKHIMOV, Singularities of Riemannian Invariants, Funct. Anal. Appl. 27 (1993), 39-50.
21. D. WAGNER, The Riemann problem in two space dimensions for a single conservation law, SIAM J. Math. Anal. 14 (1983), 534-559.
22. C. T. C. WALL, Finite determinacy of smooth map germs, Bull. London Math. Soc. 14 (1981), 481-539.
23. G. WASSERMANN, Stability of unfoldings in space and time, Acta. Math. 135 (1975), 57-128.
24. V. M. ZAKALYUKIN, Reconstructions of fronts and caustics depending on a parameter and versality of mappings, J. Sov. Math. 27 (1983), 2713-2735.