Abstract
In this paper we use the theory of compensated compactness coupled with some basic ideas of the kinetic formulation by Lions, Perthame, Souganidis & Tadmor [LPS, LPT] to establish an existence theorem for global entropy solutions of the nonstrictly hyperbolic system (1).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Caprino, S., Esposito, R., Marra, R., Pulvirenti, M.: Hydrodynamic limits of the Vlasov equation. Comm. Partial. Diff. Equations 18, 805–820 (1993)
Chueh, K.N., Conley, C.C., Smoller, J.A.: Positively invariant regions for systems of nonlinear diffusion equations. Indiana Univ. Math. J. 26, 373–411 (1977)
DiPerna, R.J.: Global solutions to a class of nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 26, 1–28 (1973)
Earnshaw, S.: On the mathematical theory of sound. Philos. Trans. 150, 1150–1154 (1858)
Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18, 95–105 (1965)
James, F., Peng, Y.J., Perthame, B.: Kinetic formulation for chromatography and some other hyperbolic systems. J. Math. Pure Appl. 74, 367–385 (1995)
Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math. 34, 481–524 (1981)
Lax, P.D.: Shock waves and entropy. In: Contributions to Nonlinear Functional Analysis, edited by E. Zarantonello. Academia Press, New York, 1971, 603–634
Lions, P.L., Perthame, B., Souganidis, P.E.: Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Comm. Pure Appl. Math. 49, 599–638 (1996)
Lions, P.L., Perthame, B., Tadmor, E.: Kinetic formulation of the isentropic gas dynamics and p-system. Commun. Math. Phys. 163, 415–431 (1994)
Lu, Y.G.: Hyperboilc Conservation Laws and the Compensated Compactness Method, Vol. 128, Chapman and Hall, CRC Press, New York, 2002
Lu, Y.G.: Convergence of the viscosity method for some nonlinear hyperbolic systems. Proc. Royal Soc. Edinburgh 124A, 341–352 (1994)
Lu, Y.G.: Convergence of the viscosity method for nonstrictly hyperbolic conservation laws. Commun. Math. Phys. 150, 59–64 (1992)
Lu, Y.G.: Some Results on General Ssystem of Isentropic Gas Dynamics. (In Russian) To appear in Diff. Equations
Murat, F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa 5, 489–507 (1978)
Oelschläger, K.: On the connection between Hamiltonian many-particle systems and the hydrodynamical equation. Arch. Ration. Mech. Anal. 115, 297–310 (1991)
Oelschläger, K.: An integro-differential equation modelling a Newtonian dynamics and its scaling limit. Arch. Ration. Mech. Anal. 137, 99–134 (1997)
Tartar, T.: Compensated compactness and applications to partial differential equations. In: Research Notes in Mathematics, Nonlinear Analysis and Mechanics, Heriot-Watt symposium Vol. 4, R. J. Knops,(ed.) Pitman Press, London, 1979
Whitham, G.B.: Linear and Nonlinear Waves. John Wiley and Sons, New York, 1973
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P.-L. Lions
Rights and permissions
About this article
Cite this article
Lu, Yg. Existence of Global Entropy Solutions of a Nonstrictly Hyperbolic System. Arch. Rational Mech. Anal. 178, 287–299 (2005). https://doi.org/10.1007/s00205-005-0379-0
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-005-0379-0