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High Amplitude Solutions for Small Data in Pairs of Conservation Laws that Change Type

In: Hyperbolic Problems: Theory, Numerics, Applications

Author

Listed:
  • V. Matos

    (Faculdade de Economia da Universidade do Porto (FEP))

  • D. Marchesin

    (Instituto Nacional de Matemática Pura e Aplicada (IMPA))

Abstract

We prove that high amplitude Riemann solutions arise from Riemann data with arbitrarily small amplitude in the hyperbolic region near the point where the rarefaction curves are tangent to the elliptic region. These solutions arise in a quadratic system of conservation laws with a compact elliptic region. The second-order terms in the fluxes correspond to type IV in Schaeffer and Shearer classification. For such Riemann data there is no small amplitude solution. Whitney perturbations of the fluxes do not change the result. Thus, we understand one possible consequence for violating strict hyperbolicity in the neighborhood of the Riemann data points. The violation of this hypothesis in Lax’s renowned theorem may yield large amplitude solutions for small data.

Suggested Citation

  • V. Matos & D. Marchesin, 2008. "High Amplitude Solutions for Small Data in Pairs of Conservation Laws that Change Type," Springer Books, in: Sylvie Benzoni-Gavage & Denis Serre (ed.), Hyperbolic Problems: Theory, Numerics, Applications, pages 711-719, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-75712-2_72
    DOI: 10.1007/978-3-540-75712-2_72
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