Abstract
We consider the 2×2 hyperbolic system of isentropic gas dynamics, in both Eulerian or Lagrangian variables (also called thep-system). We show that they can be reformulated as a kinetic equation, using an additional kinetic variable. Such a formulation was first obtained by the authors in the case of multidimensional scalar conservation laws. A new phenomenon occurs here, namely that the advection velocity is now a combination of the macroscopic and kinetic velocities. Various applications are given: we recover the invariant regions, deduce newL ∞ estimates using moments lemma and proveL ∞−w* stability for γ≥3.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen, G.Q.: The theory of compensated compactness and the System of Isentropic Gas Dynamics. Preprint MCS-P154-0590, University of Chicago, (1990)
Dafermos, C.M.: Estimates for conservation laws with little viscosity. SIAM J. Math. Anal.18, n. 2, 409–421 (1987)
DiPerna, R.J.: Convergence of Approximate Solutions to Conservation Laws. Arch. Rat. Mec. and Anal.82(1), 27–70 (1983)
DiPerna, R.J.: Convergence of the Viscosity Method of Isentropic Gas Dynamics. Commun. Math. Phys.91, 1–30 (1983)
DiPerna, R.J., Lions, P.L., Meyer, Y.:L p Regularity of Velocity Averages. Ann. I.H.P. Anal. Non Lin.8(3–4), 271–287 (1991)
Golse, F., Lions, P.L., Perthame, B., Sentis, R.: Regularity of the Moments of the Solution of a Transport Equation. J. Funct. Anal.76, 110–125 (1988)
Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock waves. CBMS-NSF regional conferences series in applied mathematics, 11 (1973)
Lions, P.L., Perthame, B.: Lemmes de moments, de moyenne et de dispersion. C.R. Acad. Sc. Paris, t.314, série I, 801–806 (1992)
Lions, P.L., Perthame, B., Tadmor, E.: Formulation cinétique des lois de conservation scalaires. C.R. Acad. Sci. Paris, t.312, série I, 97–102 (1991)
Lions, P.L., Perthame, B., Tadmor, E.: Kinetic Formulation of Scalar Conservation Laws. J.A.M.S.7, 169–191 (1994)
Murat, F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa5, 489–507 (1978)
Perthame, B.: Higher Moments Lemma: Applications to Vlasov-Poisson and Fokker-Planck equations Math. Meth. Appl. Sc.13, 441–452 (1990)
Serre, D.: Domaines invariants pour les systèmes hyperboliques de lois de conservation.69, n. 1, 46–62 (1987)
Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Berlin, Heidelberg, New York: Springer 1982
Tartar, L.: Compensated Compactness and Applications to Partial Differential Equations. In: Research Notes in Mathematics, Nonlinear analysis and mechanics, Heriot-Watt Symposium, Vol. 4 Knops, R.J. (ed.) London: Pitman Press, 1979
James, F., Peng, Y-J., Perthame, B.: Kinetic formulation for the chromatography and some other hyperbolic systems (To appear in J. Math. Pures et Appl.)
Author information
Authors and Affiliations
Additional information
Communicated by J.L. Lebowitz
Rights and permissions
About this article
Cite this article
Lions, P.L., Perthame, B. & Tadmor, E. Kinetic formulation of the isentropic gas dynamics andp-systems. Commun.Math. Phys. 163, 415–431 (1994). https://doi.org/10.1007/BF02102014
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02102014