manuscripta mathematica

, Volume 60, Issue 1, pp 49–69 | Cite as

Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system

  • Pierangelo Marcati
  • Albert J. Milani
  • Paolo Secchi


We show that the weak solutions of the nonlinear hyperbolic system
$$\left\{ \begin{gathered} \varepsilon u_t^\varepsilon + p(v^\varepsilon )_x = u^\varepsilon \hfill \\ v_t^\varepsilon - u_x^\varepsilon = 0 \hfill \\ \end{gathered} \right.$$
converge, as ε tends to zero, to the solutions of the reduced problem
$$\left\{ \begin{gathered} u + p(v)_x = 0 \hfill \\ v_t - u_x = 0 \hfill \\ \end{gathered} \right.$$
. Then they satisfy the nonlinear parabolic equation
$$v_t + p(v)_{XX} = 0$$
. The limiting procedure is carried out using the techniques of “Compensated Compactness”. Some connections with the theory of nonlinear heat conduction and the theory of nonlinear diffusion in a porous medium are suggested. The main result is stated in th. (2.9).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Aronson D., “The Porous Media Equations” to appear inSome Problems in Nonlinear Diffusion (A. Fasano, M. Primicerio Ed.) Lect. Notes in Math., SpringerGoogle Scholar
  2. [2]
    Bardos C., “Introduction aux Problèmes Hyperboliques non Linéaires”. In Lect. Notes Math. 1047, Springer Verlag, Berlin Heidelberg New York, 1984Google Scholar
  3. [3]
    Cattaneo C., “Sulla conduzione del calore”, Atti Sem. Mat. Fisico Univ. Modena 3(1948), 83–101Google Scholar
  4. [4]
    Chueh K.N. -Conley C.C. -Smoller J.A., “Positively invariant regions for systems of Nonlinear Diffusion Equations”, Indiana Univ. Mat. J. 26 (1977), 372–411Google Scholar
  5. [5]
    Dafermos C.M. -Hsiao L., “Hyperbolic Systems of balance laws with inhomogeneity and dissipation”, Indiana Univ. Math. J. 31 (1982), 471–491Google Scholar
  6. [6]
    Di Perna R J., “Convergence of Approximate Solutions to Conservation Laws”, Arch. Rat. Mech. Anal. 82 (1983), 27–70Google Scholar
  7. [7]
    Lax P.D., “Shock Waves and Entropy” inContributions to Nonlinear Functional Analysis ed. E.A. Zarantonello, Accademic Press (1971), 603–634Google Scholar
  8. [8]
    Lax P.D., “Hyperbolic Systems of Conservation Law and the Mathematical Theory of Shock Waves”, SIAM Regional Conference Serie in Math. 11 (1973)Google Scholar
  9. [9]
    Liu T.P., proceedings l'AquilaGoogle Scholar
  10. [10]
    Liu T.P., “Hyperbolic Conservation Laws with Relaxation” preprint Univ. MD College Park, July 86Google Scholar
  11. [11]
    Lions J.L., “Perturbations Singulaires dans les Problèmes aux Limites et en Controle Optimal”, Lect. Notes Math. 323, Springer-Verlag, Berlin, 1973Google Scholar
  12. [12]
    Marcati P., “Approximate Solutions to Scalar Conservation Laws via Degenerate Diffusion” inHyperbolic Equations and Related Topics (V.K. Murthy, F. Colombini Ed.) Research Notes in Mathematics, Pitman Publisher Co. (To appear)Google Scholar
  13. [13]
    Marcati P., “Approximate Solutions to Conservation laws via Convective Parabolic Equations” (submitted)Google Scholar
  14. [14]
    Milani A.J., “Long Time Existence and Singular Perturbation Results for Quasi-Linear Hyperbolic Equations with Small Parameter and Dissipation Term”. I e II. Nonlinear Analysis (to appear)Google Scholar
  15. [15]
    Oleinik, O.A., Kalashnikov A.S., Chzhou Yui-Lin, The Cauchy problem and boundary problems for equations of the type of unsteady filtration, Izv. Akad. Nauk SSSR Ser. Mat.22 (1958), 667–704Google Scholar
  16. [16]
    Sanchez-Palencia E.,Nonhomogeneous Media and Vibration theory. Lect. Notes in Phys, Springer Verlag, Berlin (198)Google Scholar
  17. [17]
    Tartar L., “Compensated Compactness and application to Partial Differential Equations”, Research Notes in Math. (R.J. Knops ed.) Pitman Press 1979Google Scholar
  18. [18]
    Vazquez J.L., “Behaviour of the velocity of one dimensional flows in Porous Media”, Trans. Amer. Mat. Soc. 286 (1984), 787–802Google Scholar
  19. [19]
    Vazquez J.L., “The hyperbolic nature of porous media equation”, to appearGoogle Scholar
  20. [20]
    Zlamal M., “Sur l'équation des Telegraphistes avec un petite Paramètre”, Atti Acc. Nazionale Lincei, Rend. Cl. Sci. Mat. Fis. Nat. 27 (1959), 324–332Google Scholar
  21. [21]
    Zlamal M., “The parabolic Equations as Limiting Case of Hyperbolic and Elliptic Equations” In Proc. Equadiff I (Prague 1962) Acad. Press; New York (1963)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Pierangelo Marcati
    • 1
  • Albert J. Milani
    • 2
  • Paolo Secchi
    • 3
  1. 1.Dipartimento di Matematica P.A.Università dell'AquilaL'AquilaItaly
  2. 2.Dipartimento di MatematicaUniversità di TorinoTorinoItaly
  3. 3.Dipartimento di MatematicaUniversità di TrentoPovo (Trento)Italy

Personalised recommendations