Convergence of Approximated Solutions to a Nonstrictly Hyperbolic System

Abstract

Abstract In this paper, four classes of special entropy-entropy flux pairs of Lax type for the nonstrictly hyperbolic system of type (1.1) (Le Roux system) are constructed based on the solutions of the standard Fuchsion equation. The second derivatives of these entropies are all singular at the point (0,O). A careful computation for these entropies at the singular point shows the compactness of η(u1, v1)t + q(u1, v1)x in H-1 loc:(R x R+) with respect to the approximated solutions constructed by using viscosity method or Friedrichs-Lax scheme method. These entropies provide a convergence theorem in the strong topology for the artifical viscosity method or Friedrichs-Lax scheme method when applied to the Cauchy problem (1.1),(1.4) and used together with the theory of compensated compactness.