THE VACUUM IN NONISENTROPIC GAS DYNAMICS

doi:10.1016/S0252-9602(12)60021-6

Abstract

Abstract:

We investigate the vacuum in nonisentropic gas dynamics in one space vari-able, with the most general equation of states allowed by thermodynamics. We recall physical constraints on the equations of state and give explicit and easily checkable condi-tions under which vacuums occur in the solution of the Riemann problem. We then present a class of models for which the Riemann problem admits unique global solutions without
vacuums.

Key words: nonisentropic gas dynamics, conservation laws, vacuum, large data, Rie-mann problem

CLC Number:

35L65

Geng Chen, Robin Young. THE VACUUM IN NONISENTROPIC GAS DYNAMICS[J].Acta mathematica scientia,Series B, 2012, 32(1): 339-351.

Trendmd

References

[1] Bressan A. Hyperbolic Systems of Conservation laws: The One-dimmensional Cauchy Problem. Oxford
Lecture Ser Math Appl 20. Oxford: Oxford Univ Press, 2000

[2] Chen Geng. Formation of singularity and smooth wave propagation for the compressible Euler equations.
to appear in J Hyp Diff Eq

[3] Chen G, Endres E, Jenssen K H. Pairwise wave interactions in ideal polytropic gases. to appear in Arch
Rat Mech Anal

[4] Chen G, Young R. Smooth solutions and singularity formation for the inhomogeneous nonlinear wave
equation. J Diff Eq, 2012, 252: 2580–2595

[5] Chen G, Young R. Shock formation and exact solutions for the compressible Euler equations with piecewise
smooth entropy. submitted.

[6] Courant R, Friedrichs K O. Supersonic Flow and Shock Waves. New York: Wiley-Interscience, 1948

[7] Dafermos C M. Hyperbolic Conservations laws in Continuum Physics. Heidelberg: Springer-Verlag, 2000

[8] Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math,
1965, 18: 697–715

[9] Lax P D. Hyperbolic systems of conservation laws, II. Comm Pure Appl Math, 1957, 10: 537–566

[10] Li T -T,Zhao Y.Vacuum problemsfor the systems of one-dimensional isentropic flow (in Chinese). Chinese
Quart J Math, 1986, 1: 41–46

[11] Lieb E. The stability of matter. Rev Mod Phys, 1976, 48: 553

[12] Liu T -P, Smoller J. On the vacuum state for the isentropic gas dynamics equations. Adv Appl Math,
1980, 1: 345–359

[13] Meniko? R, Plohr B J. The Riemann problem for fluid flow of real materials. Rev Modern Phy, 1989, 61(1): 75–130

[14] Nishida T. Global solutions for an initial boundary value problem of a quasilinear hyperbolic system. Proc
Japan Acad, 1968, 44: 642–646

[15] Smith R. The Riemann problem in gas dynamics. Trans Amer Math Soc, 1979, 249: 1–50

[16] Smoller J. Shock Waves and Reaction-di?usion Equations. New York: Springer-Verlag, 1982

[17] Temple B. Solutions in the large for the nonlinear hyperbolic conservation laws of gas dynamics. J Diff
Eqns, 1981, 41: 96–161

[18] Temple B, Young R. The large time stability of sound waves. Comm Math Phys, 1996, 179: 417–466

[19] Temple B, Young R. A paradigm for time-periodic sound wave propagation in the compressible Euler
equations. Meth Appl Anal, 2009, 16: 341–364

[20] Young R. The p-system II: The vacuum. Evolution Equations Banach Center Publications, 2003, 60: 237–252