A self-similar viscosity approach for the Riemann problem in isentropic gas dynamics and the structure of the solutions
Author:
Yong Jung Kim
Journal:
Quart. Appl. Math. 59 (2001), 637-665
MSC:
Primary 35L65; Secondary 35L67, 76N10, 76N15
DOI:
https://doi.org/10.1090/qam/1866552
MathSciNet review:
MR1866552
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Abstract: We study the Riemann problem for the system of conservation laws of one-dimensional isentropic gas dynamics in Eulerian coordinates. We construct solutions of the Riemann problem by the method of self-similar zero-viscosity limits, where the self-similar viscosity only appears in the equation for the conservation of momentum. No size restrictions on the data are imposed. The structure of the solutions obtained is also analyzed.
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C. M. Dafermos, Solution of the Riemann problem for a class of hyperbolic conservation laws by the viscosity method, Arch. Rational Mech. Analysis 52, 1–9 (1973)
C. M. Dafermos, Admissible wave fans in nonlinear hyperbolic systems, Arch. Rational Mech. Analysis 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. Akad. Nauk SSSR 127, 27–30 (1959)
T. P. Liu, The Riemann problem for qeneral $2 \times 2$ conservation laws, Trans. Amer. Math. Soc. 199, 89–112 (1974)
T. P. Liu, Admissible solutions of hyperbolic conservation laws, Memoirs Amer. Math. Soc. 240, 1–78 (1981)
P. H. Rabinowitz, Théorie de Degré Topologique et Applications à des Problèmes aux Limites non Linéaires, rédigé par H. Berestycki, Laboratoire d’Analyse Numérique, Université Paris VI, 1–300 (1975)
M. Slemrod, Resolution of the spherical piston problem for compressible isentropic gas dynamics via a self-similar viscous limit, Proc. Roy. Soc. Edinburgh Sect. A 126, 1309–1340 (1996)
M. Slemrod and A. E. Tzavaras, A limiting viscosity approach for the Riemann problem in isentropic gas dynamics, Indiana Univ. Math. J. 38, 1047–1074 (1989)
A. E. Tzavaras, Elastic as limit of viscoelastic response in a context of self-similar viscous limit, J. Diff. Equations 123, 305–341 (1995)
A. E. Tzavaras, Wave interactions and variation estimates for self-similar zero-viscosity limits in systems of conservation laws, Arch. Rational Mech. Anal. 135, 1–60 (1996)
A. I. Volpert, The spaces BV and quasilinear equations, Math. USSR Sbornik (N.S.) 73 (115), 255–302 (1967)
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© Copyright 2001
American Mathematical Society