We prove the uniqueness of Riemann solutions in the class of entropy solutions in L∞ ∩ BVloc 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 L1 ∩ L∞ ∩ BVloc 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.

The global stability of Lipschitz continuous solutions with discontinuous initial data is established in a broad class of entropy solutions in L∞ containing vacuum states. In particular, the uniqueness of Lipschitz solutions with discontinuous initial data is obtained in the broad class of entropy solutions in L∞.

We analyze global entropy solutions of the 2×2 relativistic Euler equations for isentropic fluids in special relativity and establish the uniqueness of Riemann solutions in the class of entropy solutions in L∞ ∩BVloc with arbitrarily large oscillation. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions implies their inviscid time-asymptotic stability under arbitrarily large L1 ∩ L∞ ∩ BVloc perturbation of the Riemann initial data, provided that the corresponding solutions are in L∞ and have local bounded total variation that allows the linear growth in time. This approach is also extended to deal with the stability of Riemann solutions containing vacuum in the class of entropy solutions in L∞ with arbitrarily large oscillation.

We are concerned with global entropy solutions to the relativistic Euler equations for a class of large initial data which involve the interaction of shock waves and rarefaction waves. We first carefully analyze the global behavior of the shock curves, the rarefaction wave curves, and their corresponding inverse curves in the phase plane. Based on these analyses, we use the Glimm scheme to construct global entropy solutions to the relativistic Euler equations for the class of large discontinuous initial data.

We consider in this paper the relativistic Euler equations in isentropic °uids with the equation of state p=к2p, whereк, the sound speed, is a constant less than the speed of light c. We discuss the convergence of the entropy solutions as c→∞. The analysis is based on the geometric properties of nonlinear wave curves and the Glimm's method.