A Strictly Hyperbolic System of Conservation Laws Admitting Singular Shocks

  • Herbert C. Kranzer
  • Barbara Lee Keyfitz
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 27)


The system
$$ \left\{ {\begin{array}{*{20}{c}} {{u_t} + {{\left( {{u^2} - v} \right)}_x} = 0} \\ {{v_t} + {{\left( {\frac{1}{3}{u^3} - u} \right)}_x} = 0} \\ \end{array} } \right. $$
is an example of a strictly hyperbolic, genuinely nonlinear system of conservation laws. Usually the Riemann problem for such a system is well-posed: centered weak solutions consisting of combinations of simple waves and admissible jump discontinuities (shocks) exist and are unique for each set of values of the Riemann data [1–3]. The characteristic speeds λ1 and λ2 of system (1), however, do not conform to the usual pattern for strictly hyperbolic, genuinely nonlinear systems: although locally separated, they overlap globally (cf. Keyfitz [4] for a more general discussion of the significance of overlapping characteristic speeds). In particular, λ1 = u - 1 and λ2 = u + 1 are real and unequal at any particular point U = (u, v) of state space (as strict hyperbolicity requires), and λ2 - λ1 = 2 is even bounded away from zero globally, but λ1 at one point U 1 may be equal to λ2 at a different point U 2. The corresponding right eigenvectors r 1 = (1, u + l) and r 2 = (1, u - 1) of the gradient matrix for (1) display genuine nonlinearity, since r i ∙ ▽ λi > 0 for i = 1,2 but the two eigenvalues vary in the same direction: r i ∙ ▽ λj > 0 for ij, rather than the usual “opposite variation” r i ∙ ▽ λj < 0 familiar from (say) gas dynamics. As a result, classical global existence and uniqueness theorems [3,5] no longer apply.


Rarefaction Wave Riemann Problem Characteristic Speed Viscosity Approximation Left State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Lax, P.D., Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), pp. 537–566.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Smoller, J.A., and Johnson, J.L., Global solutions for an extended class of hyperbolic systems of conservation laws, Arch. Rat. Mech. Anal., 32 (1969), pp. 169–189.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Keyfitz, B.L., and Kranzer, H.C., Existence and uniqueness of entropy solutions to the Riemann problem for hyperbolic systems of two nonlinear conservation laws, Jour. Diff. Eqns., 27 (1978), pp. 444–476.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Keyfitz, B.L., Some elementary connections among nonstrictly hyperbolic conservation laws, Contemporary Mathematics, 60 (1987), pp. 67–77.MathSciNetGoogle Scholar
  5. [5]
    Borovikov, V.A., On the decomposition of a discontinuity for a system of two quasilinear equations, Trans. Moscow Math. Soc., 27 (1972), pp. 53–94.MathSciNetMATHGoogle Scholar
  6. [6]
    Korchinski, D., Solution of the Riemann problem for a 2 × 2 system of conservation laws possessing no classical weak solution, Thesis, Adelphi University, 1977.Google Scholar
  7. [7]
    Dafermos, C.M., Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method, Arch. Rat. Mech. Anal., 52 (1973), pp. 1–9.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Dafermos, C.M., and DiPerna, R.J., The Riemann problem for certain classes of hyperbolic systems of conservation laws, Jour. Diff. Eqns., 20 (1976), pp. 90–114.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Keyfitz, B.L., and Kranzer, H.C., A viscosity approximation to a system of conservation laws with no classical Riemann solution, Proceedings of International Conference on Hyperbolic Problems, Bordeaux, 1988.Google Scholar
  10. [10]
    Shearer, M., Riemann problems for systems of nonstrictly hyperbolic conservation laws, This volume.Google Scholar
  11. [11]
    Keyfitz, B.L., and Kranzer, H.C., A system of conservation laws with no classical Riemann solution, preprint, University of Houston (1989).Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Herbert C. Kranzer
    • 1
  • Barbara Lee Keyfitz
    • 2
  1. 1.Department of Mathematics and Computer ScienceAdelphi UniversityGarden CityUSA
  2. 2.Department of MathematicsUniversity of HoustonHoustonUSA

Personalised recommendations