Nonuniqueness for the vanishing viscosity solution with fixed initial condition in a nonstrictly hyperbolic system of conservation laws

Journal of Mathematical Analysis and Applications

Volumn 335, Issue 2, 2007, Pages 996-1012

  Ostrov, Daniel N ^a  

^a SANTA CLARA UNIVERSITY   (United States)

Author keywords

Control theory; Hamiton Jacobi; Nonstrictly hyperbolic; Nonuniqueness; Viscosity

Indexed keywords

EID: 34447339954     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2007.01.102     Document Type: Article

References (21)

1. Lax P.D. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (1973), SIAM, Philadelphia
2. Bressan A., Crasta G., and Piccoli B. Well-posedness of the Cauchy problem for n × n systems of conservation laws. Mem. Amer. Math. Soc. 146 (2000)
3. Bianchini S., and Bressan A. Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. of Math. (2) 161 1 (2005) 223-342
4. Majda A., and Pego R.L. Stable viscosity matrices for systems of conservation laws. J. Differential Equations 56 2 (1985) 229-262
5. Dafermos C.M. Hyperbolic conservation laws in continuum physics. Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences) vol. 325 (2000), Springer, Berlin
6. Serre D. Systems of conservation laws II. Geometric structures, oscillations, and initial-boundary value problems (2000), Cambridge Univ. Press, Cambridge
7. Schaeffer D.G., and Shearer M. Riemann problems for nonstrictly hyperbolic 2 × 2 systems of conservation laws. Trans. Amer. Math. Soc. 304 1 (1987) 267-306
8. Keyfitz B.L., and Kranzer H.C. A system of nonstrictly hyperbolic conservation laws arising in elasticity theory. Arch. Ration. Mech. Anal. 72 3 (1979/1980) 219-241
9. Keyfitz B.L., and Kranzer H.C. The Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy. J. Differential Equations 47 1 (1983) 35-65
10. Shearer M., Schaeffer D.G., Marchesin D., and Paes-Leme P.L. Solution of the Riemann problem for a prototype 2 × 2 system of nonstrictly hyperbolic conservation laws. Arch. Ration. Mech. Anal. 97 4 (1987) 299-320
11. Freistühler H. Non-uniformity of vanishing viscosity approximation. Appl. Math. Lett. 6 2 (1993) 35-41
12. Čanić S. On the influence of viscosity on Riemann solutions. J. Dynam. Differential Equations 10 1 (1998) 109-149
13. Freistühler H. Instability of vanishing viscosity approximation to hyperbolic systems of conservation laws with rotational invariance. J. Differential Equations 87 2 (1990) 205-226
14. Freistühler H. Rotational degeneracy of hyperbolic systems of conservation laws. Arch. Ration. Mech. Anal. 113 1 (1990) 39-64
15. Freistühler H. Dynamical stability and vanishing viscosity: A case study of a non-strictly hyperbolic system. Comm. Pure Appl. Math. 45 5 (1992) 561-582
16. Freistühler H., and Liu T.-P. Nonlinear stability of overcompressive shock waves in a rotationally invariant system of viscous conservation laws. Comm. Math. Phys. 153 1 (1993) 147-158
17. Azevedo A.V., and Marchesin D. Multiple viscous solutions for systems of conservation laws. Trans. Amer. Math. Soc. 347 8 (1995) 3061-3077
18. Azevedo A.V., Marchesin D., Plohr B.J., and Zumbrun K. Nonuniqueness of solutions of Riemann problems. Z. Angew. Math. Phys. 47 6 (1996) 977-998
19. Isaacson E., Marchesin D., Plohr B., and Temple B. The Riemann problem near a hyperbolic singularity: The classification of solutions of quadratic Riemann problems I. SIAM J. Appl. Math. 48 5 (1988) 1009-1032
20. LeVeque R.J. Numerical Methods for Conservation Laws. Lectures Math. ETH Zürich. second ed. (1992), Birkhäuser, Basel
21. Fleming W.H., and Soner H.M. Controlled Markov Processes and Viscosity Solutions. Appl. Math. (NY) vol. 25 (1993), Springer, New York