References
R.Courant and K. O.Friedrichs, Supersonic flow and shock waves. New York 1948.
H. Freistühler, Rotational degeneracy of hyperbolic systems of conservation laws. Arch. Rational Mech. Anal.113, 39–64 (1991).
H. Freistühler, Dynamical stability and vanishing viscosity: a case study of a nonstrictly hyperbolic system. Comm. Pure Appl. Math.45, 561–582 (1992).
H. Freistühler, Non-uniformity of vanishing viscosity approximation. Appl. Math. Lett.6, 35–41 (1993).
H. Freistühler, On the stability of non-classical shock waves. Habilitationsschrift, RWTH Aachen, 1994.
H. Freistühler, On the persistence of ideal shock waves. Appl. Math. Lett.7, 7–11 (1994).
H. Freistühler andT.-P. Liu, Nonlinear stability of overcompressive shock waves in a rotationally invariant system of viscous conservation laws. Comm. Math. Phys.153, 147–158 (1993).
H.Freistühler and P.Szmolyan, Existence and bifurcation of viscous profiles for all intermediate magnetonydrodynamic shock waves. SIAM J. Math. Anal.26 (1995).
E. Isaacson, D. Marchesin, andB. Plohr, Transitional waves for conservation laws. SIAM J. Math. Anal.21, 837–866 (1990).
A.Jeffrey and T.Taniuti, Nonlinear wave propagation. New York-London 1964.
P. Lax, Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math.10, 537–566 (1957).
P.Lax, Hyperbolic systems of conservation laws in several space variables. In: Current topics in p. d. e. Y. Ohya, K. Kasahara, and N. Shimakura, eds., Tokyo 1986.
T.-T.Li, private communication, July 1991.
T.-T.Li and W.-C.Yu, Boundary value problems for quasilinear hyperbolic systems. Duke University 1985.
T.-P. Liu, The Riemann problem for general systems of conservation laws. J. Differential Equations18, 218–234 (1975).
T.-P.Liu and K.Zumbrun, Stability of an undercompressive shock. Preprint.
A.Majda, Existence of multidimensional shock fronts. Amer. Math. Soc. Mem.281 (1983).
A. Majda andR. Pego, Stable viscosity matrices for systems of conservation laws. J. Differential Equations56, 229–262 (1985).
D. G. Schaeffer, M. Shearer, D. Marchesin, andP. J. Paes-Leme, Solution of the Riemann problem for a prototype system of non-strictly hyperbolic conservation laws. Arch. Rational Mech. Anal.97, 299–320 (1987).
S.Schecter and M.Shearer, Riemann problems involving undercompressive shocks. In: Partial differential equations and continuum models of phase transitions, M. Rascle, D. Serre, and M. Slemrod, eds., Lecture Notes in Physics344, 187–200. Berlin-Heidelberg-New York 1989.
S.Schecter and M.Shearer, Transversality for undercompressive shocks in Riemano problems. In: Viscous profiles and numerical methods for shock waves, M. Shearer, ed., Philadelphia 1991.
K.Zumbrun, private communication, May 1993.
K. Zumbrun, B. Plohr, andD. Marchesin, Scattering behavior of transitional shock waves. Mat. Contemp.3, 191–209 (1992).
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Freistühler, H. A short note on the persistence of ideal shock waves. Arch. Math 64, 344–352 (1995). https://doi.org/10.1007/BF01198091
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DOI: https://doi.org/10.1007/BF01198091