The Riemann problem in gas dynamics

Author:
Randolph G. Smith

Journal:
Trans. Amer. Math. Soc. **249** (1979), 1-50

MSC:
Primary 35L67; Secondary 35Q20, 76J99, 76L05, 76M05

DOI:
https://doi.org/10.1090/S0002-9947-1979-0526309-2

MathSciNet review:
526309

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Abstract: We consider the Riemann problem (R.P.) for the system of gas dynamics equations in a single space variable. We assume that the specific internal energy (*s* = specific entropy, *v* = specific volume) satisfies the usual hypotheses, pressure); we also assume some reasonable hypotheses about the asymptotic behavior of *e*. We call functions *e* satisfying these hypotheses *energy functions*

Theorem 1. *For any initial data* , , *u = flow velocity), the R. P. has a solution*.

We introduce two conditions:

Theorem 2. (I) *is necessary and sufficient for uniqueness of solutions of the R. P. Nonuniqueness persists under small perturbations of the initial data*.

(I) is implied by the known condition

() |

which holds for all usual gases. (I) implies (II). We construct energy functions

*e*that violate (II), that satisfy (II) but violate (I), and that satisfy (I) but violate (*).

In all solutions considered, the shocks satisfy the entropy condition and the Lax shock conditions.

**[1]**R. Courant and K. O. Friedrichs,*Supersonic Flow and Shock Waves*, Interscience Publishers, Inc., New York, N. Y., 1948. MR**0029615****[2]**W. D. Hayes,*Gasdynamic discontinuities*, Princeton Univ. Press, Princeton, N. J., 1960.**[3]**P. D. Lax,*Hyperbolic systems of conservation laws. II*, Comm. Pure Appl. Math.**10**(1957), 537–566. MR**0093653**, https://doi.org/10.1002/cpa.3160100406**[4]**Tai Ping Liu,*Shock waves in the nonisentropic gas flow*, J. Differential Equations**22**(1976), no. 2, 442–452. MR**0417589**, https://doi.org/10.1016/0022-0396(76)90039-5**[5]**J. A. Smoller,*On the solution of the Riemann problem with general step data for an extended class of hyperbolic systems*, Michigan Math. J.**16**(1969), 201–210. MR**0247283**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1979-0526309-2

Keywords:
Conservation law,
Riemann problem,
gas dynamics,
discontinuous data,
compressible fluid flow,
Hugoniot curve,
shocks,
rarefaction waves,
contact discontinuity,
ideal gases,
Lax shock conditions,
supersonic flow,
equation of state,
nonhomentropic gas flow

Article copyright:
© Copyright 1979
American Mathematical Society