Hostname: page-component-669899f699-2mbcq Total loading time: 0 Render date: 2025-05-02T16:22:17.729Z Has data issue: false hasContentIssue false
  • English
  • Français

Resolution of the spherical piston problem for compressible isentropic gas dynamics via a self-similar viscous limit

Published online by Cambridge University Press:  14 November 2011

M. Slemrod
M. Slemrod
Affiliation:
Center for the Mathematical Sciences, University of Wisconsin-Madison, Madison, WI 53715-1149, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper proves the existence of solutions to the spherical piston problem for isentropic gas dynamics with equation of state p(p) = Apγ, γ≧ 1. The method of analysis is to replace the usual viscosity ε with εt, thus permitting a search for self-similar viscous limits. The main result of the paper is that self-similar viscous limits are proved to exist and converge to a solution to the piston problem when

N = 1, 2, 3 space dimensions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 6 , 1996 , pp. 1309 - 1340
Copyright
Copyright © Royal Society of Edinburgh 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Chen, G.-Q.. Remarks on global solutions to the compressible Euler equations with spherical symmetry (Preprint).Google Scholar
2Chen, G.-Q. and Glimm, J.. Global solutions to the compressible Euler equations with geometrical structure (Preprint).Google Scholar
3Courant, R. and Friedrichs, K. O.. Supersonic Flow and Shock Waves, Applied Mathematical Sciences 21 (Berlin: Springer, 1976).CrossRefGoogle Scholar
4Dafermos, C. M.. Solution of the Riemann problem for a class of hyperbolic conservation laws by the viscosity method. Arch. Rational Mech. Anal. 52 (1973), 1–9.CrossRefGoogle Scholar
5Dafermos, C. M.. Structure of solutions of the Riemann problem for hyperbolic systems of conservation laws. Arch. Rational Mech. Anal. 53 (1974), 203–17.CrossRefGoogle Scholar
6Dafermos, C. M.. Admissible wave fans in nonlinear hyperbolic systems. Arch. Rational Mech. Anal. 106 (1989), 243–60.CrossRefGoogle Scholar
7Dafermos, C. M. and DiPerna, R. J.. The Riemann problem for certain classes of hyperbolic systems of conservation laws. J. Differential Equations 20 (1976), 90–114.CrossRefGoogle Scholar
8Fan, H. T.. A limiting ‘viscosity approach’ to the Riemann problem for materials exhibiting change of phase (II). Arch. Rational Mech. Anal. 116 (1992), 317–38.CrossRefGoogle Scholar
9Fan, H. T.. Self-similar solutions for a modified Broadwell model and its fluid dynamic limits. SIAM J. Math. Anal, (to appear).Google Scholar
10Hoff, D.. Spherically symmetric solutions of the Navier–Stokes equations for compressible, isothermal flow with large, discontinuous initial data. Indiana Univ. Math. J. 41 (1992), 1225–302.CrossRefGoogle Scholar
11Hoff, D.. Global solutions of the Navier–Stokes equations for multidimensional compressible flow with discontinuous initial data (Preprint 9306, Institute for Applied Mathematics and Scientific Computation, Indiana University, Bloomington, 1993).Google Scholar
12Kalasnikov, A. S.. 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 (1957), 27–30 (in Russian).Google Scholar
13Keyfitz, B. and Kranzer, H.. A viscosity approximation to a system of conservation laws with no classical Riemann solutions. In Proceedings of International Conference on Nonlinear Hyperbolic Problems pp. 185–97, eds Carasso, C., Charrier, I., Hanouzet, B., Joly, J. L., Springer Lecture Notes in Mathematics, (Springer-Verlag, Berlin-Heidelberg, 1989).CrossRefGoogle Scholar
14Lions, P.-L.. Existence globale de solutions pour les equations de Navier-Stokes compressible isentropiques(Preprint 9317, Ceremade, University of Paris IX Dauphine, 1993).Google Scholar
15Makino, T., Mizohata, K. and Ukai, S.. The global weak solutions of compressible Euler equations with spherical symmetry. Japan J. Indust. Appl. Math. 9 (1992), 431–49.CrossRefGoogle Scholar
16Makino, T., Mizohata, K. and Ukai, S.. Global weak solutions of the compressible Euler equations with spherical symmetry (II). Res. Rep. Inf. Sci, Tokyo Inst. Tech., A–III (1993).Google Scholar
17Rabinowitz, P. H.. Théorie de Degré Topologique et Applications à des Problèmes aux Limites non Lineaires, ed. Berestycki, H. (Paris: Laboratoire d'Analyse Numerique, Universite Paris VI, 1975).Google Scholar
18Sedov, L. I.. Similarity and Dimensional Methods in Mechanics, 4th edn (Moscow: GITTL, 1957; English translation: New York, Academic Press, 1959).Google Scholar
19Slemrod, M.. A limiting ‘viscosity’ approach to the Riemann problem for materials exhibiting change of phase. Arch. Rational Mech. Anal. 105 (1989), 327–65.CrossRefGoogle Scholar
20Slemrod, M.. A comparison of two viscous regularizations of the Riemann problem for Burgers' equation. SIAM J. Math. Anal, (to appear).Google Scholar
21Slemrod, M. and Tzavaras, A. E.. A limiting viscosity approach for the Riemann problem for isentropic gas dynamics. Indiana Univ. Math. J. 38 (1989), 1047–74.CrossRefGoogle Scholar
22Slemrod, M. and Tzavaras, A. E.. Self-similar fluid dynamic limits for the Broadwell system. Arch. Rational Mech. Anal. 122 (1993), 353–92.CrossRefGoogle Scholar
23Taylor, G. I.. The air wave surrounding an expanding sphere. Proc. Roy. Sec. London Ser. A 186 (1946), 273–92.Google ScholarPubMed
24Tupciev, V. A.. Collapse of an arbitrary discontinuity for a system of two quasilinear first order equations. Z. Vycisl. Mat. i Fiz. 4 (1964), 817–25; English translation: USSR Comput. Math, and Math. Physics (5) 4 (1964), 36–48.Google Scholar
25Tupciev, V. A.. On the method of introducing viscosity in the study of problems involving decay of a discontinuity. Dokl. Acad. Nauk SSSR 211 (1973), 55–8; English translation: Soviet Math. Dokl. 14 (1973), 978–82.Google Scholar
26Tzavaras, A. E.. Elastic as a limit of viscoelastic response, in a context of self-similar viscous limits. J. Differential Equations (to appear).Google Scholar
27Tzavaras, A. E.. Wave structure induced by the fluid dynamic limits in the Broadwell model. Arch. Rational Mech. Anal, (submitted).Google Scholar