Communications in Mathematical Physics

, Volume 228, Issue 2, pp 201–217

Uniqueness and Stability of Riemann Solutions¶with Large Oscillation in Gas Dynamics


  • Gui-Qiang Chen
    • Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA. E-mail:
  • Hermano Frid
    • Instituto de Matemática Pura e Aplicada, Est. Dona Castorina, 110, Jardim Botânico, Cep 22460-320,¶Rio de Janeiro, RJ, Brazil. E-mail:
  • Yachun Li
    • Department of Mathematics, Shanghai Jiaotong University, Shanghai 200030, P.R. China.E-mail:

DOI: 10.1007/s002200200615

Cite this article as:
Chen, G., Frid, H. & Li, Y. Commun. Math. Phys. (2002) 228: 201. doi:10.1007/s002200200615


We prove the uniqueness of Riemann solutions in the class of entropy solutions in \(\) with arbitrarily large oscillation for the 3 × 3 system of Euler equations in gas dynamics. The proof for solutions with large oscillation is based on a detailed analysis of the global behavior of shock curves in the phase space and the singularity of centered rarefaction waves near the center in the physical plane. The uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily large\(\) perturbation of the Riemann initial data, as long as the corresponding solutions are in L and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is needed. The uniqueness result for Riemann solutions can easily be extended to entropy solutions U(x,t), piecewise Lipschitz in x, for any t > 0, with arbitrarily large oscillation.

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© Springer-Verlag Berlin Heidelberg 2002