Decay of solutions for the equations of isothermal gas dynamics

The equations of isothermal gas dynamics are expressed as a 2×2-system of genuinely nonlinear hyperbolic conservation laws which possesses a convex entropy. Existence of weak global solutions is known for even large initial data via Glimm’s difference scheme. We show that the total variation of such (large) solutions decays strongly to zero. The proof consists in showing that the total amount of wave interaction in the Glimm approximations is uniformly bounded.

[1]  Richard Courant,et al.  Supersonic Flow And Shock Waves , 1948 .

[2]  P. Lax.  Hyperbolic systems of conservation laws II , 1957 .

[3]  J. Glimm.  Solutions in the large for nonlinear hyperbolic systems of equations , 1965 .

[4]  Takaaki Nishida,et al.  Global solution for an initial boundary value problem of a quasilinear hyperbolic system , 1968 .

[5]  P. Lax,et al.  Decay of solutions of systems of nonlinear hyperbolic conservation laws , 1970 .

[6]  P. Lax.  Shock Waves and Entropy , 1971 .

[7]  P. Lax,et al.  Systems of conservation equations with a convex extension. , 1971, Proceedings of the National Academy of Sciences of the United States of America.

[8]  Tai-Ping Liu.  Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws , 1977 .

[9]  Tai-Ping Liu.  Large-time behavior of solutions of initial and initial-boundary value problems of a general system of hyperbolic conservation laws , 1977 .

[10]  R. J. Diperna,et al.  Decay of solutions of hyperbolic systems of conservation laws with a convex extension , 1977 .

[11]  Tai-Ping Liu.  The deterministic version of the Glimm scheme , 1977 .

[12]  Tai-Ping Liu.  Decay to N-waves of solutions of general systems of nonlinear hyperbolic conservation laws , 1977 .

[13]  西田 孝明,et al.  Nonlinear hyperbolic equations and related topics in fluid dynamics , 1978 .

[14]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[15]  Kurt Friedrichs,et al.  [71-1] Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. USA, 68 (1971), 1686–1688 , 1986 .