The structure of solutions of gas dynamic equations and the formation of the vacuum state
Author:
Hai Tao Fan
Journal:
Quart. Appl. Math. 49 (1991), 29-48
MSC:
Primary 76N15
DOI:
https://doi.org/10.1090/qam/1096230
MathSciNet review:
MR1096230
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Abstract: The structure of the solutions of the isentropic gas dynamics equations in Eulerian coordinates given by [17] is studied. A condition under which the formation of the vacuum state occurs is obtained.
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R. DiPerna, Existence in the large nonlinear hyperbolic conservation laws, Arch. Rational Mech. Anal. 52, 244–257 (1973)
J. Greenberg, On the interaction of shocks and simple waves of the same family, Arch. Rational Mech. Anal. 37, 136–160 (1970)
T.-P. Liu, Shock waves in the nonisentropic gas flow, J. Differential Equations 22, 442–452 (1976)
T.-P. Liu, Solution in the large for the equations of nonisentropic gas dynamics, Indiana Univ. Math. J. 26, 147–177 (1977)
T.-P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57, 135–148 (1977)
T. Nishida and J. Smoller, Solutions in the large for some nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math. 26, 183–200 (1973)
R. Smith, The Riemann problem in gas dynamics, Trans. Amer. Math. Soc. 249, 1–30 (1979)
Zhang Tong and Guo Yu-Fa, A class of initial value problems for systems of aerodynamic equations, Acta. Math. Sinica 15 386–396 (1965), English translation in Chinese Math. 7, 90–101 (1965)
T.-P. Liu and J. Smoller, On the vacuum state for the isentropic gas dynamics equations, Advances in Appl. Math. 1, 345–359 (1980)
C. M. Dafermos, Solution of the Riemann problem for a class of hyperbolic conservation laws by the viscosity method, Arch. Rational Mech. Anal. 52, 1–9 (1973)
C. M. Dafermos, Structure of the solutions of the Riemann problem for hyperbolic conservation laws, Arch. Rational Mech. Anal. 53, 203–217 (1974)
C. M. Dafermos and R. J. DiPerna, The Riemann problem for certain classes of hyperbolic systems of conservation laws, J. Differential Equations 20, 90–114 (1976)
C. M. Dafermos, Admissible wave fans in nonlinear hyperbolic system, Arch. Rational Mech. Anal. 106, 243–260 (1989)
A. S. Kalasnikov, Construction of generalized solutions of quasi-linear equations of first order without convexity conditions as limits of solutions of parabolic equations with a small parameter, Dokl. Adak. Nauk. SSSR 127, 27–30 (1959)
B. L. Keyfitz and H. C. Kranzer, A viscosity approximation to a system of conservation laws with no classical Riemann solution, to appear in Proc. of Int. Conf. on Hyperbolic Problems, Bordeanux, (1989)
M. Slemrod, A limiting “viscosity” approach to the Riemann problem for materials exhibiting change of phase, Arch. Rational Mech. Anal. 105, 327–365 (1989)
M. Slemrod and A. Tzavaras, A limiting viscosity approach for the Riemann problem in isentropic gas dynamics, preprint (1989)
V. A. Tupciev, On the method of introducing viscosity in the study of problems involving the decay of discontinuity, Dokl. Akad. Nauk. SSSR 211, 55–58 (1973)
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© Copyright 1991
American Mathematical Society
